3.3 Övningar
Sommarmatte 1
(Skillnad mellan versioner)
| Versionen från 29 april 2007 kl. 12.53 (redigera) Ossiang (Diskussion | bidrag) (→Övning 3.3:1) ← Gå till föregående ändring |
Nuvarande version (31 maj 2007 kl. 07.02) (redigera) (ogör) KTH.SE:u1rp004j (Diskussion | bidrag) |
||
| (14 mellanliggande versioner visas inte.) | |||
| Rad 1: | Rad 1: | ||
| + | |||
| ==Övning 3.3:1== | ==Övning 3.3:1== | ||
| <div class="ovning"> | <div class="ovning"> | ||
| - | Bestäm x om | + | Bestäm $\,x\,$ om |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| - | <tr><td height="5px"/></tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext">$10^x=1\,000$</td> | + | <td class="ntext" width="50%">$10^x=1\,000$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext">$10^x=0,1$</td> | + | <td class="ntext" width="50%">$10^x=0{,}1$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext">$\displaystyle \frac{1}{10^x}=100$</td> | + | <td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=100$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext">$\displaystyle \frac{1}{10^x}=0,000\,1$</td> | + | <td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=0{,}000\,1$</td> |
| </tr> | </tr> | ||
| <tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
| Rad 21: | Rad 23: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| - | <tr><td height="5px"/></tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext">$x=3$</td> | + | <td class="ntext" width="50%">$x=3$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext">$x=-1$</td> | + | <td class="ntext" width="50%">$x=-1$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext">$x=-2$</td> | + | <td class="ntext" width="50%">$x=-2$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext">$x=4$</td> | + | <td class="ntext" width="50%">$x=4$</td> |
| </tr> | </tr> | ||
| <tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
| Rad 40: | Rad 43: | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga a<br \>[[Bild:3_3_1a.gif]] | + | Lösning till delfråga a) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_1a.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| Rad 48: | Rad 58: | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b </div> | + | <div class=NavHead>Lösning b) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga b<br \>[[Bild:3_3_1b.gif]] | + | Lösning till delfråga b) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_1b.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| Rad 56: | Rad 73: | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning c) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga c<br \>[[Bild:3_3_1c.gif]] | + | Lösning till delfråga c) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_1c.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| Rad 64: | Rad 88: | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning d </div> | + | <div class=NavHead>Lösning d) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga d<br \>[[Bild:3_3_1d.gif]] | + | Lösning till delfråga d) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_1d.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| Rad 75: | Rad 106: | ||
| Beräkna | Beräkna | ||
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| - | <tr><td height="5px"/></tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext">$\lg{ 0,1}$</td> | + | <td class="ntext" width="25%">$\lg{ 0{,}1}$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext">$\lg{ 10\,000}$</td> | + | <td class="ntext" width="25%">$\lg{ 10\,000}$</td> |
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext">$\lg {0,001}$</td> | + | <td class="ntext" width="25%">$\lg {0{,}001}$</td> |
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext">$\lg {1}$</td> | + | <td class="ntext" width="25%">$\lg {1}$</td> |
| </tr> | </tr> | ||
| - | <tr><td height="5px"/></tr> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext">$10^{\lg{2}}$</td> | + | <td class="ntext" width="25%">$10^{\lg{2}}$</td> |
| <td class="ntext">f)</td> | <td class="ntext">f)</td> | ||
| - | <td class="ntext">$\lg{10^3}$</td> | + | <td class="ntext" width="25%">$\lg{10^3}$</td> |
| <td class="ntext">g)</td> | <td class="ntext">g)</td> | ||
| - | <td class="ntext">$10^{-\lg{0,1}}$</td> | + | <td class="ntext" width="25%">$10^{-\lg{0{,}1}}$</td> |
| <td class="ntext">h)</td> | <td class="ntext">h)</td> | ||
| - | <td class="ntext">$\lg{\displaystyle \frac{1}{10^2}}$</td> | + | <td class="ntext" width="25%">$\lg{\displaystyle \frac{1}{10^2}}$</td> |
| </tr> | </tr> | ||
| <tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
| </table> | </table> | ||
| - | |||
| </div> | </div> | ||
| Rad 105: | Rad 133: | ||
| <div class=NavHead>Facit </div> | <div class=NavHead>Facit </div> | ||
| <div class=NavContent> | <div class=NavContent> | ||
| - | Facit till alla delfrågorna<br \> | + | Facit till alla delfrågor |
| <table width="100%" cellspacing="10px"> | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="25%">$-1$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="25%">$4$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="25%">$-3$</td> | ||
| + | <td class="ntext">d)</td> | ||
| + | <td class="ntext" width="25%">$0$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">e)</td> | ||
| + | <td class="ntext" width="25%">$2$</td> | ||
| + | <td class="ntext">f)</td> | ||
| + | <td class="ntext" width="25%">$3$</td> | ||
| + | <td class="ntext">g)</td> | ||
| + | <td class="ntext" width="25%">$10$</td> | ||
| + | <td class="ntext">h)</td> | ||
| + | <td class="ntext" width="25%">$-2$</td> | ||
| + | </tr> | ||
| <tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning a) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga a) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2a.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning b) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga b) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2b.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning c) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga c) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2c.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning d) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga d) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2d.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning e) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga e) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2e.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning f) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga f) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2f.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning g) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga g) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2g.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning h) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga h) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_2h.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | ==Övning 3.3:3== | ||
| + | |||
| + | <div class="ovning"> | ||
| + | Beräkna | ||
| + | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| <td class="ntext">a)</td> | <td class="ntext">a)</td> | ||
| - | <td class="ntext">$-1$</td> | + | <td class="ntext" width="33%">$\log_2{8}$</td> |
| <td class="ntext">b)</td> | <td class="ntext">b)</td> | ||
| - | <td class="ntext">$4$</td> | + | <td class="ntext" width="33%">$\log_9{\displaystyle \frac{1}{3}}$</td> |
| <td class="ntext">c)</td> | <td class="ntext">c)</td> | ||
| - | <td class="ntext">$-3$</td> | + | <td class="ntext" width="33%">$\log_2{0{,}125}$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">d)</td> | <td class="ntext">d)</td> | ||
| - | <td class="ntext">$0$</td> | + | <td class="ntext" width="33%">$\log_3{\left(9\cdot3^{1/3}\right)}$</td> |
| + | <td class="ntext">e)</td> | ||
| + | <td class="ntext" width="33%">$2^{\log_{\scriptstyle2}{4}}$</td> | ||
| + | <td class="ntext">f)</td> | ||
| + | <td class="ntext" width="33%">$\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">g)</td> | ||
| + | <td class="ntext" width="33%">$\log_3{12}-\log_3{4}$</td> | ||
| + | <td class="ntext">h)</td> | ||
| + | <td class="ntext" width="33%">$\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$</td> | ||
| </tr> | </tr> | ||
| <tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Facit </div> | ||
| + | <div class=NavContent> | ||
| + | Facit till alla delfrågor | ||
| + | <table width="100%" cellspacing="10px"> | ||
| <tr align="left"> | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$3$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$-\displaystyle \frac{1}{2}$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$-3$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">d)</td> | ||
| + | <td class="ntext" width="33%">$\displaystyle \frac{7}{3}$</td> | ||
| <td class="ntext">e)</td> | <td class="ntext">e)</td> | ||
| - | <td class="ntext">$2$</td> | + | <td class="ntext" width="33%">$4$</td> |
| <td class="ntext">f)</td> | <td class="ntext">f)</td> | ||
| - | <td class="ntext">$3$</td> | + | <td class="ntext" width="33%">$-2$</td> |
| + | </tr> | ||
| + | <tr align="left"> | ||
| <td class="ntext">g)</td> | <td class="ntext">g)</td> | ||
| - | <td class="ntext">$10$</td> | + | <td class="ntext" width="33%">$1$</td> |
| <td class="ntext">h)</td> | <td class="ntext">h)</td> | ||
| - | <td class="ntext">$-2$</td> | + | <td class="ntext" width="33%">$\displaystyle \frac{5}{2}$</td> |
| </tr> | </tr> | ||
| <tr><td height="5px"/></tr> | <tr><td height="5px"/></tr> | ||
| Rad 135: | Rad 336: | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning a </div> | + | <div class=NavHead>Lösning a) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga a<br \>[[Bild:3_3_2a.gif]] | + | Lösning till delfråga a) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3a.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning b) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga b) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3b.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning b </div> | + | <div class=NavHead>Lösning c) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga b<br \>[[Bild:3_3_2b.gif]] | + | Lösning till delfråga c) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3c.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning d) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga d) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3d.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning c </div> | + | <div class=NavHead>Lösning e) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga c<br \>[[Bild:3_3_2c.gif]] | + | Lösning till delfråga e) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3e.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning f) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga f) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3f.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning d </div> | + | <div class=NavHead>Lösning g) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga d<br \>[[Bild:3_3_2d.gif]] | + | Lösning till delfråga g) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3g.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning h) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga h) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_3h.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | ==Övning 3.3:4== | ||
| + | |||
| + | <div class="ovning"> | ||
| + | Förenkla | ||
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$\lg{50}-\lg{5}$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$\lg{23}+\lg{\displaystyle \frac{1}{23}}$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$</td> | ||
| + | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| + | </table> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning e </div> | + | <div class=NavHead>Facit </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga e<br \>[[Bild:3_3_2e.gif]] | + | Facit till alla delfrågor |
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$1$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$0$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$-\displaystyle \frac{1}{2}\lg{3}$</td> | ||
| + | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning a) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga a) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_4a.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning f </div> | + | <div class=NavHead>Lösning b) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga f<br \>[[Bild:3_3_2f.gif]] | + | Lösning till delfråga b) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_4b.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning c) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga c) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_4c.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | ==Övning 3.3:5== | ||
| + | |||
| + | <div class="ovning"> | ||
| + | Förenkla | ||
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$\ln{e^3}+\ln{e^2}$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$\ln{8}-\ln{4}-\ln{2}$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$(\ln{1})\cdot e^2$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">d)</td> | ||
| + | <td class="ntext" width="33%">$\ln{e}-1$</td> | ||
| + | <td class="ntext">e)</td> | ||
| + | <td class="ntext" width="33%">$\ln{\displaystyle \frac{1}{e^2}}$</td> | ||
| + | <td class="ntext">f)</td> | ||
| + | <td class="ntext" width="33%">$\left(e^{\ln{e}}\right)^2$</td> | ||
| + | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| + | </table> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning g </div> | + | <div class=NavHead>Facit </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga g<br \>[[Bild:3_3_2g.gif]] | + | Facit till alla delfrågor |
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="33%">$5$</td> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="33%">$0$</td> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="33%">$0$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">d)</td> | ||
| + | <td class="ntext" width="33%">$0$</td> | ||
| + | <td class="ntext">e)</td> | ||
| + | <td class="ntext" width="33%">$-2$</td> | ||
| + | <td class="ntext">f)</td> | ||
| + | <td class="ntext" width="33%">$e^2$</td> | ||
| + | </tr> | ||
| + | <tr><td height="5px"/></tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning a) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga a) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_5a.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| <div class=NavFrame style="CLEAR: both"> | <div class=NavFrame style="CLEAR: both"> | ||
| - | <div class=NavHead>Lösning h </div> | + | <div class=NavHead>Lösning b) </div> |
| <div class=NavContent> | <div class=NavContent> | ||
| - | Lösning till delfråga h<br \>[[Bild:3_3_2h.gif]] | + | Lösning till delfråga b) |
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_5b.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning c) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga c) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_5c.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning d) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga d) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_5d.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning e) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga e) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_5e.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning f) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga f) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_5f.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | ==Övning 3.3:6== | ||
| + | |||
| + | <div class="ovning"> | ||
| + | [[Bild:miniraknare.gif||right]] | ||
| + | Använd miniräknaren till höger för att beräkna med tre decimaler (Knappen <tt>LN</tt> betecknar den naturliga logaritmen i basen ''e''): | ||
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"><td height="5px"/></tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="100%">$\log_3{4}$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="100%">$\lg{46}$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="100%">$\log_3{\log_2{(3^{118})}}$</td> | ||
| + | </tr> | ||
| + | <tr align="left"><td height="40px"/></tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Facit </div> | ||
| + | <div class=NavContent> | ||
| + | Facit till alla delfrågor | ||
| + | <table width="100%" cellspacing="10px"> | ||
| + | <tr align="left"><td height="5px"/></tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">a)</td> | ||
| + | <td class="ntext" width="100%">$1{,}262$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">b)</td> | ||
| + | <td class="ntext" width="100%">$1{,}663$</td> | ||
| + | </tr> | ||
| + | <tr align="left"> | ||
| + | <td class="ntext">c)</td> | ||
| + | <td class="ntext" width="100%">$4{,}762$</td> | ||
| + | </tr> | ||
| + | <tr align="left"><td height="5px"/></tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning a) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga a) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_6a-1(2).gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_6a-2(2).gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning b) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga b) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_6b.gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class=NavFrame style="CLEAR: both"> | ||
| + | <div class=NavHead>Lösning c) </div> | ||
| + | <div class=NavContent> | ||
| + | Lösning till delfråga c) | ||
| + | <table width="100%"> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_6c-1(2).gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td align="center"> | ||
| + | [[Bild:3_3_6c-2(2).gif]] | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| </div> | </div> | ||
| </div> | </div> | ||
Nuvarande version
Innehåll |
[redigera] Övning 3.3:1
Bestäm $\,x\,$ om
| a) | $10^x=1\,000$ | b) | $10^x=0{,}1$ |
| c) | $\displaystyle \frac{1}{10^x}=100$ | d) | $\displaystyle \frac{1}{10^x}=0{,}000\,1$ |
Facit
Facit till alla delfrågor
| a) | $x=3$ | b) | $x=-1$ |
| c) | $x=-2$ | d) | $x=4$ |
Lösning a)
Lösning till delfråga a)
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Lösning b)
Lösning till delfråga b)
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Lösning c)
Lösning till delfråga c)
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Lösning d)
Lösning till delfråga d)
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[redigera] Övning 3.3:2
Beräkna
| a) | $\lg{ 0{,}1}$ | b) | $\lg{ 10\,000}$ | c) | $\lg {0{,}001}$ | d) | $\lg {1}$ |
| e) | $10^{\lg{2}}$ | f) | $\lg{10^3}$ | g) | $10^{-\lg{0{,}1}}$ | h) | $\lg{\displaystyle \frac{1}{10^2}}$ |
Facit
Facit till alla delfrågor
| a) | $-1$ | b) | $4$ | c) | $-3$ | d) | $0$ |
| e) | $2$ | f) | $3$ | g) | $10$ | h) | $-2$ |
Lösning a)
Lösning till delfråga a)
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Lösning b)
Lösning till delfråga b)
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Lösning c)
Lösning till delfråga c)
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Lösning d)
Lösning till delfråga d)
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Lösning e)
Lösning till delfråga e)
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Lösning f)
Lösning till delfråga f)
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Lösning g)
Lösning till delfråga g)
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Lösning h)
Lösning till delfråga h)
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[redigera] Övning 3.3:3
Beräkna
| a) | $\log_2{8}$ | b) | $\log_9{\displaystyle \frac{1}{3}}$ | c) | $\log_2{0{,}125}$ |
| d) | $\log_3{\left(9\cdot3^{1/3}\right)}$ | e) | $2^{\log_{\scriptstyle2}{4}}$ | f) | $\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$ |
| g) | $\log_3{12}-\log_3{4}$ | h) | $\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$ | ||
Facit
Facit till alla delfrågor
| a) | $3$ | b) | $-\displaystyle \frac{1}{2}$ | c) | $-3$ |
| d) | $\displaystyle \frac{7}{3}$ | e) | $4$ | f) | $-2$ |
| g) | $1$ | h) | $\displaystyle \frac{5}{2}$ | ||
Lösning a)
Lösning till delfråga a)
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Lösning b)
Lösning till delfråga b)
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Lösning c)
Lösning till delfråga c)
|
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Lösning d)
Lösning till delfråga d)
|
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Lösning e)
Lösning till delfråga e)
|
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Lösning f)
Lösning till delfråga f)
|
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Lösning g)
Lösning till delfråga g)
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Lösning h)
Lösning till delfråga h)
|
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[redigera] Övning 3.3:4
Förenkla
| a) | $\lg{50}-\lg{5}$ | b) | $\lg{23}+\lg{\displaystyle \frac{1}{23}}$ | c) | $\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$ |
Facit
Facit till alla delfrågor
| a) | $1$ | b) | $0$ | c) | $-\displaystyle \frac{1}{2}\lg{3}$ |
Lösning a)
Lösning till delfråga a)
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Lösning b)
Lösning till delfråga b)
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Lösning c)
Lösning till delfråga c)
|
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[redigera] Övning 3.3:5
Förenkla
| a) | $\ln{e^3}+\ln{e^2}$ | b) | $\ln{8}-\ln{4}-\ln{2}$ | c) | $(\ln{1})\cdot e^2$ |
| d) | $\ln{e}-1$ | e) | $\ln{\displaystyle \frac{1}{e^2}}$ | f) | $\left(e^{\ln{e}}\right)^2$ |
Facit
Facit till alla delfrågor
| a) | $5$ | b) | $0$ | c) | $0$ |
| d) | $0$ | e) | $-2$ | f) | $e^2$ |
Lösning a)
Lösning till delfråga a)
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Lösning b)
Lösning till delfråga b)
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Lösning c)
Lösning till delfråga c)
|
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Lösning d)
Lösning till delfråga d)
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Lösning e)
Lösning till delfråga e)
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Lösning f)
Lösning till delfråga f)
|
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[redigera] Övning 3.3:6
Använd miniräknaren till höger för att beräkna med tre decimaler (Knappen LN betecknar den naturliga logaritmen i basen e):
| a) | $\log_3{4}$ |
| b) | $\lg{46}$ |
| c) | $\log_3{\log_2{(3^{118})}}$ |
Facit
Facit till alla delfrågor
| a) | $1{,}262$ |
| b) | $1{,}663$ |
| c) | $4{,}762$ |
Lösning a)
Lösning till delfråga a)
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Lösning b)
Lösning till delfråga b)
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Lösning c)
Lösning till delfråga c)
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