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>
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+<div class="svar">
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 <table width="100%" cellspacing="10px">
 <tr><td height="5px"/></tr>
 <tr><td height="5px"/></tr>
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-</div>
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-<div class=NavFrame style="CLEAR: both">
+==&Ouml;vning 3.2:2==
-<div class=NavHead>L&ouml;sning &nbsp;</div>
-<div class=NavContent>
+<div class="ovning">
 <table width="100%" cellspacing="10px">
-<tr>
+<tr><td height="5px"/></tr>
-<td align="center">
+<tr align="left">
-[[Bild:3_2_1-1(3).gif]]
+<td class="ntext" width="100%">L&ouml;s ekvationen \ \sqrt{2x+7}=x+2\,.
 </td>
 </tr>
-<tr>
+<tr><td height="5px"/></tr>
-<td align="center">
+</table>
-[[Bild:3_2_1-2(3).gif]]
+</div>
-</td>
+<div class="svar">
+<table width="100%" cellspacing="10px">
+<tr><td height="5px"/></tr>
+<tr align="left">
+<td class="ntext" width="100%">x=1</td>
 </tr>
-<tr>
+<tr><td height="5px"/></tr>
-<td align="center">
+</table>
-[[Bild:3_2_1-3(3).gif]]
+</div>
-</td>
+==&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="svar">
+<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>
+</tr>
+<tr><td height="5px"/></tr>
+</table>
 </div>
-==&Ouml;vning 3.2:2==
+==&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 $\ \sqrt{2x+7}=x+2\,$.
+<td class="ntext" width="100%">L&ouml;s ekvationen $\ \sqrt{1-x}=2-x\,$.</td>
-</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%">x=1</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_2-1(3).gif]]
+<td class="ntext" width="100%">x=1</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 3.2:6==
 <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 $\ \sqrt{x+1}+\sqrt{x+5}=4\,$.</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%">$\left \{ \eqalign{ x_1 & = 3 \cr x_2 & = 4\cr } \right.$</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_3-1(2).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>
+<tr align="left">
-<td align="center">
+<td class="ntext">c)</td>
-[[Bild:3_2_3-2(2).gif]]
+<td class="ntext" width="50%">$x=-2$</td>
-</td>
+<td class="ntext">d)</td>
+<td class="ntext" width="50%">x=4</td>
 </tr>
+<tr><td height="5px"/></tr>
 </table>
-</div>
 </div>
-==&Ouml;vning 3.2:4==
+==&Ouml;vning 3.3:2==
 <div class="ovning">
+Ber&auml;kna
 <table width="100%" cellspacing="10px">
-<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">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>
 </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 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 3.3:3==
+<div class="ovning">
+Ber&auml;kna
+<table width="100%" cellspacing="10px">
 <tr align="left">
-<td class="ntext" width="100%">Saknar l&ouml;sning.</td>
+<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>
-<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="33%">3</td>
-</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>
 </table>
-</div>
 </div>
+==&Ouml;vning 3.3:4==
-==&Ouml;vning 3.2:5==
 <div class="ovning">
+F&ouml;renkla
 <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>
+<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>
 </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 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>
+==&Ouml;vning 3.3:5==
+<div class="ovning">
+F&ouml;renkla
+<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%">$\ln{e^3}+\ln{e^2}$</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>
-<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>