Övningar 3.3

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Versionen från 16 juli 2007 kl. 08.09 (redigera)
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(Ny sida: __NOTOC__ ==&Ouml;vning 3.3:1== <div class="ovning"> Best&auml;m $\,x\,$ om <table width="100%" cellspacing="10px"> <tr align="left"> <td class="ntext">a)</td> <td class="ntext" width="50%"...)
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Rad 1: Rad 1:
__NOTOC__ __NOTOC__
-==&Ouml;vning 3.3:1==+'''&Ouml;vning 3.3:1'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Best&auml;m $\,x\,$ om Best&auml;m $\,x\,$ om
-<table width="100%" cellspacing="10px">+<table width="100%" style="margin-top:-20px; margin-bottom:-5px;">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$10^x=1\,000$</td> <td class="ntext" width="50%">$10^x=1\,000$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$10^x=0{,}1$</td> <td class="ntext" width="50%">$10^x=0{,}1$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=100$</td> <td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=100$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=0{,}000\,1$</td> <td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=0{,}000\,1$</td>
</tr> </tr>
Rad 20: Rad 20:
</div> </div>
-==&Ouml;vning 3.3:2==+'''&Ouml;vning 3.3:2'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Ber&auml;kna Ber&auml;kna
-<table width="100%" cellspacing="10px">+<table width="100%" style="margin-top:-20px; margin-bottom:-5px;">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$\lg{ 0{,}1}$</td> <td class="ntext" width="25%">$\lg{ 0{,}1}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$\lg{ 10\,000}$</td> <td class="ntext" width="25%">$\lg{ 10\,000}$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$\lg {0{,}001}$</td> <td class="ntext" width="25%">$\lg {0{,}001}$</td>
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$\lg {1}$</td> <td class="ntext" width="25%">$\lg {1}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$10^{\lg{2}}$</td> <td class="ntext" width="25%">$10^{\lg{2}}$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$\lg{10^3}$</td> <td class="ntext" width="25%">$\lg{10^3}$</td>
-<td class="ntext">g)</td>+<td class="ntext">g)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$10^{-\lg{0{,}1}}$</td> <td class="ntext" width="25%">$10^{-\lg{0{,}1}}$</td>
-<td class="ntext">h)</td>+<td class="ntext">h)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="25%">$\lg{\displaystyle \frac{1}{10^2}}$</td> <td class="ntext" width="25%">$\lg{\displaystyle \frac{1}{10^2}}$</td>
</tr> </tr>
Rad 48: Rad 48:
</div> </div>
-==&Ouml;vning 3.3:3==+'''&Ouml;vning 3.3:3'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
Ber&auml;kna Ber&auml;kna
-<table width="100%" cellspacing="10px">+<table width="100%" style="margin-top:-20px; margin-bottom:-5px;">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_2{8}$</td> <td class="ntext" width="33%">$\log_2{8}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_9{\displaystyle \frac{1}{3}}$</td> <td class="ntext" width="33%">$\log_9{\displaystyle \frac{1}{3}}$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_2{0{,}125}$</td> <td class="ntext" width="33%">$\log_2{0{,}125}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_3{\left(9\cdot3^{1/3}\right)}$</td> <td class="ntext" width="33%">$\log_3{\left(9\cdot3^{1/3}\right)}$</td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$2^{\log_{\scriptstyle2}{4}}$</td> <td class="ntext" width="33%">$2^{\log_{\scriptstyle2}{4}}$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$</td> <td class="ntext" width="33%">$\log_2{4}+\log_2{\displaystyle \frac{1}{16}}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">g)</td>+<td class="ntext">g)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_3{12}-\log_3{4}$</td> <td class="ntext" width="33%">$\log_3{12}-\log_3{4}$</td>
-<td class="ntext">h)</td>+<td class="ntext">h)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$</td> <td class="ntext" width="33%">$\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$</td>
</tr> </tr>
Rad 78: Rad 78:
</div> </div>
-==&Ouml;vning 3.3:4==+'''&Ouml;vning 3.3:4'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
F&ouml;renkla F&ouml;renkla
-<table width="100%" cellspacing="10px">+<table width="100%" style="margin-top:-20px; margin-bottom:-5px;">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\lg{50}-\lg{5}$</td> <td class="ntext" width="33%">$\lg{50}-\lg{5}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\lg{23}+\lg{\displaystyle \frac{1}{23}}$</td> <td class="ntext" width="33%">$\lg{23}+\lg{\displaystyle \frac{1}{23}}$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$</td> <td class="ntext" width="33%">$\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$</td>
</tr> </tr>
Rad 94: Rad 94:
</div> </div>
-==&Ouml;vning 3.3:5==+'''&Ouml;vning 3.3:5'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
F&ouml;renkla F&ouml;renkla
-<table width="100%" cellspacing="10px">+<table width="100%" style="margin-top:-20px; margin-bottom:-5px;">
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\ln{e^3}+\ln{e^2}$</td> <td class="ntext" width="33%">$\ln{e^3}+\ln{e^2}$</td>
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\ln{8}-\ln{4}-\ln{2}$</td> <td class="ntext" width="33%">$\ln{8}-\ln{4}-\ln{2}$</td>
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$(\ln{1})\cdot e^2$</td> <td class="ntext" width="33%">$(\ln{1})\cdot e^2$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">d)</td>+<td class="ntext">d)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\ln{e}-1$</td> <td class="ntext" width="33%">$\ln{e}-1$</td>
-<td class="ntext">e)</td>+<td class="ntext">e)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\ln{\displaystyle \frac{1}{e^2}}$</td> <td class="ntext" width="33%">$\ln{\displaystyle \frac{1}{e^2}}$</td>
-<td class="ntext">f)</td>+<td class="ntext">f)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="33%">$\left(e^{\ln{e}}\right)^2$</td> <td class="ntext" width="33%">$\left(e^{\ln{e}}\right)^2$</td>
</tr> </tr>
Rad 118: Rad 118:
</div> </div>
-==&Ouml;vning 3.3:6==+'''&Ouml;vning 3.3:6'''
-<div class="ovning">+<div class="ovning" style="margin-top:-20px; margin-bottom:-5px;">
[[Bild:miniraknare.gif||right]] [[Bild:miniraknare.gif||right]]
Anv&auml;nd minir&auml;knaren till h&ouml;ger f&ouml;r att ber&auml;kna med tre decimaler (Knappen <tt>LN</tt> betecknar den naturliga logaritmen i basen ''e''): Anv&auml;nd minir&auml;knaren till h&ouml;ger f&ouml;r att ber&auml;kna med tre decimaler (Knappen <tt>LN</tt> betecknar den naturliga logaritmen i basen ''e''):
-<table width="100%" cellspacing="10px">+<table width="100%" >
<tr align="left"><td height="5px"/></tr> <tr align="left"><td height="5px"/></tr>
<tr align="left"> <tr align="left">
-<td class="ntext">a)</td>+<td class="ntext">a)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">$\log_3{4}$</td> <td class="ntext" width="100%">$\log_3{4}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">b)</td>+<td class="ntext">b)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">$\lg{46}$</td> <td class="ntext" width="100%">$\lg{46}$</td>
</tr> </tr>
<tr align="left"> <tr align="left">
-<td class="ntext">c)</td>+<td class="ntext">c)&nbsp;&nbsp;&nbsp;</td>
<td class="ntext" width="100%">$\log_3{\log_2{(3^{118})}}$</td> <td class="ntext" width="100%">$\log_3{\log_2{(3^{118})}}$</td>
</tr> </tr>

Versionen från 16 juli 2007 kl. 11.26

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)    $\displaystyle \frac{1}{10^x}=0{,}000\,1$

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)    $\lg{\displaystyle \frac{1}{10^2}}$

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)    $\log_a{\bigl(a^2\sqrt{a}\,\bigr)}$

Övning 3.3:4

Förenkla

a)    $\lg{50}-\lg{5}$ b)    $\lg{23}+\lg{\displaystyle \frac{1}{23}}$ c)    $\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}$

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)    $\left(e^{\ln{e}}\right)^2$

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)    $\log_3{\log_2{(3^{118})}}$
Den här artikeln är hämtad från http://wiki.math.se/wikis/sf0600_0701/index.php/%C3%96vningar_3.3
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