Övn 3

Sommarmatte 1

(Skillnad mellan versioner)

Nuvarande version

[redigera] Övning 3.1:1

Skriv i potensform

a)	$\sqrt{2}$	b)	$\sqrt{7^5}$	c)	$\bigl(\sqrt[\scriptstyle3]{3}\,\bigr)^4$	d)	$\sqrt{\sqrt{3}}$

a)	$2^{1/2}$	b)	$7^{5/2}$	c)	$3^{4/3}$	d)	$3^{1/4}$

[redigera] Övning 3.1:2

Förenkla så långt som möjligt

a)	$\sqrt{3^2}$	b)	$\sqrt{\left(-3\right)^2}$	c)	$\sqrt{-3^2}$	d)	$\sqrt{5}\cdot\sqrt[\scriptstyle3]{5}\cdot5$
e)	$\sqrt{18}\cdot\sqrt{8}$	f)	$\sqrt[\scriptstyle3]{8}$	g)	$\sqrt[\scriptstyle3]{-125}$

a)	$3$	b)	$3$	c)	ej definierad	d)	$5^{11/6}$
e)	$12$	f)	$2$	g)	$-5$

[redigera] Övning 3.1:3

Förenkla så långt som möjligt

a)	$\bigl(\sqrt{5}-\sqrt{2}\,\bigr)\bigl(\sqrt{5}+\sqrt{2}\,\bigr)$	b)	$\displaystyle \frac{\sqrt{96}}{\sqrt{18}}$
c)	$\sqrt{16+\sqrt{16}}$	d)	$\sqrt{\displaystyle \frac{2}{3}}\bigl(\sqrt{6}-\sqrt{3}\,\bigr)$

a)	$3$	b)	$\displaystyle \frac{4\sqrt{3}}{3}$
c)	$2\sqrt{5}$	d)	$2-\sqrt{2}$

[redigera] Övning 3.1:4

Förenkla så långt som möjligt

a)	$\sqrt{0{,}16}$	b)	$\sqrt[\scriptstyle3]{0{,}027}$
c)	$\sqrt{50}+4\sqrt{20}-3\sqrt{18}-2\sqrt{80}$	d)	$\sqrt{48}+ \sqrt{12} +\sqrt{3} -\sqrt{75}$

a)	$0{,}4$	b)	$0{,}3$
c)	$-4\sqrt{2}$	d)	$2\sqrt{3}$

[redigera] Övning 3.1:5

Skriv som ett uttryck utan rottecken i nämnaren.

a)	$\displaystyle \frac{2}{\sqrt{12}}$	b)	$\displaystyle \frac{1}{\sqrt[\scriptstyle3]{7}}$	c)	$\displaystyle \frac{2}{3+\sqrt{7}}$	d)	$\displaystyle \frac{1}{\sqrt{17}-\sqrt{13}}$

a)	$\displaystyle \frac{\sqrt{3}}{3}$	b)	$\displaystyle \frac{7^{2/3}}{7}$	c)	$3-\sqrt{7}$	d)	$\displaystyle \frac{\sqrt{17}+\sqrt{13}}{4}$

[redigera] Övning 3.1:6

Skriv som ett uttryck utan rottecken i nämnaren.

a)	$\displaystyle \frac{\sqrt{2}+3}{\sqrt{5}-2}$	b)	$\displaystyle \frac{1}{\left(\sqrt{3}-2\right)^2-2}$
c)	$\displaystyle \frac{\displaystyle \frac{1}{\sqrt{3}}-\displaystyle \frac{1}{\sqrt{5}}}{\displaystyle \frac{1}{\sqrt{2}}-\displaystyle \frac{1}{2}}$	d)	$\displaystyle \frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{6}}$

a)	$6+2\sqrt{2}+3\sqrt{5}+\sqrt{10}$	b)	$-\displaystyle \frac{5+4\sqrt{3}}{23}$
c)	$\displaystyle \frac{2}{3}\sqrt{6}+\displaystyle \frac{2}{3}\sqrt{3}-\displaystyle \frac{2}{5}\sqrt{10}-\displaystyle \frac{2}{5}\sqrt{5}$	d)	$\displaystyle \frac{5\sqrt{3}+7\sqrt{2}-\sqrt{6}-12}{23}$

[redigera] Övning 3.1:7

Förenkla så långt som möjligt

a)	$\displaystyle \frac{1}{\sqrt{6}-\sqrt{5}} - \displaystyle \frac{1}{\sqrt{7}-\sqrt{6}}$	b)	$\displaystyle \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}$	c)	$\displaystyle \sqrt{153}-\sqrt{68}$

a)	$\sqrt{5}-\sqrt{7}$	b)	$-\sqrt{35}$	c)	$\sqrt{17}$

[redigera] Övning 3.1:8

Avgör vilket tal som är störst av

a)	$\sqrt[\scriptstyle3]5\ $ och $\ \sqrt[\scriptstyle3]6$	b)	$\sqrt7\ $ och $\ 7$
c)	$\sqrt7\ $ och $\ 2{,}5$	d)	$\sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3\ $ och $\ \sqrt[\scriptstyle3]2\cdot3$

a)	$\sqrt[\scriptstyle3]6 > \sqrt[\scriptstyle3]5$	b)	$7 > \sqrt7$
c)	$\sqrt7 > 2{,}5$	d)	$\sqrt[\scriptstyle3]2\cdot3 > \sqrt2\bigl(\sqrt[\scriptstyle4]3\,\bigr)^3$

[redigera] Övning 3.2:1

Lös ekvationen $\ \sqrt{x-4}=6-x\,$.

$x=5$

[redigera] Övning 3.2:2

Lös ekvationen $\ \sqrt{2x+7}=x+2\,$.

$x=1$

[redigera] Övning 3.2:3

Lös ekvationen $\ \sqrt{3x-8}+2=x\,$.

$\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$

[redigera] Övning 3.2:4

Lös ekvationen $\ \sqrt{1-x}=2-x\,$.

Saknar lösning.

[redigera] Övning 3.2:5

Lös ekvationen $\ \sqrt{3x-2}=2-x\,$.

$x=1$

[redigera] Övning 3.2:6

Lös ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$.

$x=\displaystyle\frac{5}{4}$

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

a)	$x=3$	b)	$x=-1$
c)	$x=-2$	d)	$x=4$

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

a)	$-1$	b)	$4$	c)	$-3$	d)	$0$
e)	$2$	f)	$3$	g)	$10$	h)	$-2$

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

a)	$3$	b)	$-\displaystyle \frac{1}{2}$	c)	$-3$
d)	$\displaystyle \frac{7}{3}$	e)	$4$	f)	$-2$
g)	$1$	h)	$\displaystyle \frac{5}{2}$

[redigera] Övning 3.3:4

Förenkla

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

a)	$1$	b)	$0$	c)	$-\displaystyle \frac{1}{2}\lg{3}$

[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)	$\left(e^{\ln{e}}\right)^2$

a)	$5$	b)	$0$	c)	$0$
d)	$0$	e)	$-2$	f)	$e^2$

[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)	$\log_3{\log_2{(3^{118})}}$


a)	$1{,}262$
b)	$1{,}663$
c)	$4{,}762$

[redigera] Övning 3.4:1

Lös ekvationerna

a)	$e^x=13$	b)	$13e^x=2\cdot3^{-x}$	c)	$3e^x=7\cdot2^x$

a)	$x=\ln 13$	b)	$x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$	c)	$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$

[redigera] Övning 3.4:2

Lös ekvationerna

a)	$2^{\scriptstyle x^2-2}=1$	b)	$e^{2x}+e^x=4$	c)	$3e^{x^2}=2^x$

a)	$ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $	b)	$x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$	c)	Saknar lösning

[redigera] Övning 3.4:3

Lös ekvationerna

a)	$2^{-x^2}=2e^{2x}$	b)	$\ln{(x^2+3x)}=\ln{(3x^2-2x)}$
c)	$\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$

a)	$x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}$	b)	$x=\displaystyle \frac{5}{2}$
c)	$x=1$

Den här artikeln är hämtad från http://wiki.math.se/wikis/sf0600_0701/index.php/%C3%96vn_3

Övn 3

Sommarmatte 1

Nuvarande version

[redigera] Övning 3.1:1

[redigera] Övning 3.1:2

[redigera] Övning 3.1:3

[redigera] Övning 3.1:4

[redigera] Övning 3.1:5

[redigera] Övning 3.1:6

[redigera] Övning 3.1:7

[redigera] Övning 3.1:8

[redigera] Övning 3.2:1

[redigera] Övning 3.2:2

[redigera] Övning 3.2:3

[redigera] Övning 3.2:4

[redigera] Övning 3.2:5

[redigera] Övning 3.2:6

[redigera] Övning 3.3:1

[redigera] Övning 3.3:2

[redigera] Övning 3.3:3

[redigera] Övning 3.3:4

[redigera] Övning 3.3:5

[redigera] Övning 3.3:6

[redigera] Övning 3.4:1

[redigera] Övning 3.4:2

[redigera] Övning 3.4:3

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 </div>
-<div class=NavFrame style="CLEAR: both">
+<div class="svar">
-<div class=NavHead>Facit&nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
 <tr><td height="5px"/></tr>
 </table>
 </div>
+==&Ouml;vning 3.2:3==
+<div class="ovning">
+<table width="100%" cellspacing="10px">
+<tr><td height="5px"/></tr>
+<tr align="left">
+<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{3x-8}+2=x\,$.</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
 </div>
-<div class=NavFrame style="CLEAR: both">
+<div class="svar">
-<div class=NavHead>L&ouml;sning &nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
-<tr>
+<tr><td height="5px"/></tr>
-<td align="center">
+<tr align="left">
-[[Bild:3_2_2-1(3).gif]]
+<td class="ntext" width="100%">$\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$</td>
-</td>
-</tr>
-<tr>
-<td align="center">
-[[Bild:3_2_2-2(3).gif]]
-</td>
-</tr>
-<tr>
-<td align="center">
-[[Bild:3_2_2-3(3).gif]]
-</td>
 </tr>
+<tr><td height="5px"/></tr>
 </table>
-</div>
 </div>
-==&Ouml;vning 3.2:3==
+==&Ouml;vning 3.2:4==
 <div class="ovning">
 <tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{3x-8}+2=x\,$.</td>
+<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{1-x}=2-x\,$.</td>
 </tr>
 <tr><td height="5px"/></tr>
 </div>
-<div class=NavFrame style="CLEAR: both">
+<div class="svar">
-<div class=NavHead>Facit&nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
 <tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext" width="100%">$\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$</td>
+<td class="ntext" width="100%">Saknar l&ouml;sning.</td>
 </tr>
 <tr><td height="5px"/></tr>
 </table>
 </div>
+==&Ouml;vning 3.2:5==
+<div class="ovning">
+<table width="100%" cellspacing="10px">
+<tr><td height="5px"/></tr>
+<tr align="left">
+<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{3x-2}=2-x\,$.</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
 </div>
-<div class=NavFrame style="CLEAR: both">
+<div class="svar">
-<div class=NavHead>L&ouml;sning &nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
-<tr>
+<tr><td height="5px"/></tr>
-<td align="center">
+<tr align="left">
-[[Bild:3_2_3-1(2).gif]]
+<td class="ntext" width="100%">$x=1$</td>
-</td>
-</tr>
-<tr>
-<td align="center">
-[[Bild:3_2_3-2(2).gif]]
-</td>
 </tr>
+<tr><td height="5px"/></tr>
 </table>
-</div>
 </div>
-==&Ouml;vning 3.2:4==
+==&Ouml;vning 3.2:6==
 <div class="ovning">
 <tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{1-x}=2-x\,$.</td>
+<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$.</td>
 </tr>
 <tr><td height="5px"/></tr>
 </div>
-<div class=NavFrame style="CLEAR: both">
+<div class="svar">
-<div class=NavHead>Facit&nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
 <tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext" width="100%">Saknar l&ouml;sning.</td>
+<td class="ntext" width="100%">$x=\displaystyle\frac{5}{4}$</td>
 </tr>
 <tr><td height="5px"/></tr>
 </table>
 </div>
+==&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%">$10^x=1\,000$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$10^x=0{,}1$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=100$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$\displaystyle \frac{1}{10^x}=0{,}000\,1$</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
 </div>
-<div class=NavFrame style="CLEAR: both">
+<div class="svar">
-<div class=NavHead>L&ouml;sning &nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
-<tr>
+<tr align="left">
-<td align="center">
+<td class="ntext">a)</td>
-[[Bild:3_2_4.gif]]
+<td class="ntext" width="50%">$x=3$</td>
-</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$x=-1$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$x=-2$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">$x=4$</td>
 </tr>
+<tr><td height="5px"/></tr>
 </table>
 </div>
+==&Ouml;vning 3.3:2==
+<div class="ovning">
+Ber&auml;kna
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="25%">$\lg{ 0{,}1}$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="25%">$\lg{ 10\,000}$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="25%">$\lg {0{,}001}$</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="25%">$\lg {1}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">e)</td>
+<td class="ntext" width="25%">$10^{\lg{2}}$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="25%">$\lg{10^3}$</td>
+<td class="ntext">g)</td>
+<td class="ntext" width="25%">$10^{-\lg{0{,}1}}$</td>
+<td class="ntext">h)</td>
+<td class="ntext" width="25%">$\lg{\displaystyle \frac{1}{10^2}}$</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
 </div>
+<div class="svar">
+<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>
+</table>
+</div>
-==&Ouml;vning 3.2:5==
+==&Ouml;vning 3.3:3==
 <div class="ovning">
+Ber&auml;kna
 <table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$\log_2{8}$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$\log_9{\displaystyle \frac{1}{3}}$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$\log_2{0{,}125}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">d)</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><td height="5px"/></tr>
+</table>
+</div>
+<div class="svar">
+<table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{3x-2}=2-x\,$.</td>
+<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" width="33%">$4$</td>
+<td class="ntext">f)</td>
+<td class="ntext" width="33%">$-2$</td>
+</tr>
+<tr align="left">
+<td class="ntext">g)</td>
+<td class="ntext" width="33%">$1$</td>
+<td class="ntext">h)</td>
+<td class="ntext" width="33%">$\displaystyle \frac{5}{2}$</td>
 </tr>
 <tr><td height="5px"/></tr>
 </div>
-<div class=NavFrame style="CLEAR: both">
+==&Ouml;vning 3.3:4==
-<div class=NavHead>Facit&nbsp;</div>
-<div class=NavContent>
+<div class="ovning">
+F&ouml;renkla
 <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="svar">
+<table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext" width="100%">$x=1$</td>
+<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>
+==&Ouml;vning 3.3:5==
+<div class="ovning">
+F&ouml;renkla
+<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="svar">
-<div class=NavHead>L&ouml;sning &nbsp;</div>
-<div class=NavContent>
 <table width="100%" cellspacing="10px">
-<tr>
+<tr align="left">
-<td align="center">
+<td class="ntext">a)</td>
-[[Bild:3_2_5-1(2).gif]]
+<td class="ntext" width="33%">$5$</td>
-</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>
+<tr align="left">
-<td align="center">
+<td class="ntext">d)</td>
-[[Bild:3_2_5-2(2).gif]]
+<td class="ntext" width="33%">$0$</td>
-</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>
+==&Ouml;vning 3.3:6==
+<div class="ovning">
+[[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''):
+<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>
-==&Ouml;vning 3.2:6==
+<div class="svar">
+<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>
+==&Ouml;vning 3.4:1==
 <div class="ovning">
+L&ouml;s ekvationerna
 <table width="100%" cellspacing="10px">
-<tr><td height="5px"/></tr>
 <tr align="left">
-<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{x+1}+\sqrt{x+5}=4\,$.</td>
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$e^x=13$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$13e^x=2\cdot3^{-x}$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$3e^x=7\cdot2^x$</td>
+<tr><td height="5px"/></tr>
 </tr>
+</table>
+</div>
+<div class="svar">
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$x=\ln 13$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$x=\displaystyle\frac{\ln 2 - \ln 13}{1+\ln 3}$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$x=\displaystyle\frac{\ln 7 - \ln 3}{1-\ln 2}$</td>
 <tr><td height="5px"/></tr>
+</tr>
 </table>
 </div>
-<div class=NavFrame style="CLEAR: both">
-<div class=NavHead>Facit&nbsp;</div>
+==&Ouml;vning 3.4:2==
-<div class=NavContent>
+<div class="ovning">
+L&ouml;s ekvationerna
 <table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$2^{\scriptstyle x^2-2}=1$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$e^{2x}+e^x=4$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">$3e^{x^2}=2^x$</td>
+</tr>
 <tr><td height="5px"/></tr>
+</table>
+</div>
+<div class="svar">
+<table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext" width="100%">$x=\displaystyle\frac{5}{4}$</td>
+<td class="ntext">a)</td>
+<td class="ntext" width="33%">$ \left\{ \eqalign{ x_1&=\sqrt2 \cr x_2&=-\sqrt2 } \right. $</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="33%">$x=\ln \left(\displaystyle\frac{\sqrt17}{2}-\frac{1}{2}\right)$</td>
+<td class="ntext">c)</td>
+<td class="ntext" width="33%">Saknar l&ouml;sning</td>
 </tr>
 <tr><td height="5px"/></tr>
 </table>
-</div>
 </div>
-<div class=NavFrame style="CLEAR: both">
+==&Ouml;vning 3.4:3==
-<div class=NavHead>L&ouml;sning &nbsp;</div>
-<div class=NavContent>
+<div class="ovning">
+L&ouml;s ekvationerna
 <table width="100%" cellspacing="10px">
-<tr>
+<tr align="left">
-<td align="center">
+<td class="ntext">a)</td>
-[[Bild:3_2_6-1(2).gif]]
+<td class="ntext" width="50%">$2^{-x^2}=2e^{2x}$</td>
-</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$\ln{(x^2+3x)}=\ln{(3x^2-2x)}$</td>
 </tr>
-<tr>
+<tr align="left">
-<td align="center">
+<td class="ntext">c)</td>
-[[Bild:3_2_6-2(2).gif]]
+<td class="ntext" width="50%">$\ln{x}+\ln{(x+4)}=\ln{(2x+3)}$</td>
-</td>
 </tr>
+<tr><td height="5px"/></tr>
 </table>
 </div>
+<div class="svar">
+<table width="100%" cellspacing="10px">
+<tr align="left">
+<td class="ntext">a)</td>
+<td class="ntext" width="50%">$x=-\,\displaystyle\frac{1}{\ln{2}}\pm\sqrt{\left(\displaystyle\frac{1}{\ln{2}}\right)^2-1}$</td>
+<td class="ntext">b)</td>
+<td class="ntext" width="50%">$x=\displaystyle \frac{5}{2}$</td>
+</tr>
+<tr align="left">
+<td class="ntext">c)</td>
+<td class="ntext" width="50%">$x=1$</td>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
 </div>