3.3 Exercises

From Förberedande kurs i matematik 1

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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
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{{Mall:Ej vald flik|[[3.3 Logaritmer|Teori]]}}
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{{Not selected tab|[[3.3 Logarithms|Theory]]}}
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{{Mall:Vald flik|[[3.3 Övningar|Övningar]]}}
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{{Selected tab|[[3.3 Exercises|Exercises]]}}
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===Övning 3.3:1===
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===Exercise 3.3:1===
<div class="ovning">
<div class="ovning">
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Best&auml;m <math>\,x\,</math> om
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Solve the following equations for <math>x</math>.
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
|width="50%" | <math>10^x=1\,000</math>
|width="50%" | <math>10^x=1\,000</math>
|b)
|b)
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|width="50%" | <math>10^x=0{,}1</math>
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|width="50%" | <math>10^x=0\textrm{.}1</math>
|-
|-
|c)
|c)
|width="50%" | <math>\displaystyle \frac{1}{10^x}=100</math>
|width="50%" | <math>\displaystyle \frac{1}{10^x}=100</math>
|d)
|d)
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|width="50%" | <math>\displaystyle \frac{1}{10^x}=0{,}000\,1</math>
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|width="50%" | <math>\displaystyle \frac{1}{10^x}=0\textrm{.}000\,1</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:1|Lösning a|Lösning 3.3:1a|Lösning b|Lösning 3.3:1b|Lösning c|Lösning 3.3:1c|Lösning d|Lösning 3.3:1d}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:1|Solution a|Solution 3.3:1a|Solution b|Solution 3.3:1b|Solution c|Solution 3.3:1c|Solution d|Solution 3.3:1d}}
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===Övning 3.3:2===
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===Exercise 3.3:2===
<div class="ovning">
<div class="ovning">
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Ber&auml;kna
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Calculate
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="25%" | <math>\lg{ 0{,}1}</math>
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|width="25%" | <math>\lg{ 0\textrm{.}1}</math>
|b)
|b)
|width="25%" | <math>\lg{ 10\,000}</math>
|width="25%" | <math>\lg{ 10\,000}</math>
|c)
|c)
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|width="25%" | <math>\lg {0{,}001}</math>
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|width="25%" | <math>\lg {0\textrm{.}001}</math>
|d)
|d)
|width="25%" | <math>\lg {1}</math>
|width="25%" | <math>\lg {1}</math>
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|width="25%" | <math>\lg{10^3}</math>
|width="25%" | <math>\lg{10^3}</math>
|g)
|g)
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|width="25%" | <math>10^{-\lg{0{,}1}}</math>
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|width="25%" | <math>10^{-\lg{0\textrm{.}1}}</math>
|h)
|h)
|width="25%" | <math>\lg{\displaystyle \frac{1}{10^2}}</math>
|width="25%" | <math>\lg{\displaystyle \frac{1}{10^2}}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:2|Lösning a|Lösning 3.3:2a|Lösning b|Lösning 3.3:2b|Lösning c|Lösning 3.3:2c|Lösning d|Lösning 3.3:2d|Lösning e|Lösning 3.3:2e|Lösning f|Lösning 3.3:2f|Lösning g|Lösning 3.3:2g|Lösning h|Lösning 3.3:2h}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:2|Solution a|Solution 3.3:2a|Solution b|Solution 3.3:2b|Solution c|Solution 3.3:2c|Solution d|Solution 3.3:2d|Solution e|Solution 3.3:2e|Solution f|Solution 3.3:2f|Solution g|Solution 3.3:2g|Solution h|Solution 3.3:2h}}
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===Övning 3.3:3===
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===Exercise 3.3:3===
<div class="ovning">
<div class="ovning">
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Ber&auml;kna
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Calculate
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%" | <math>\log_9{\displaystyle \frac{1}{3}}</math>
|width="33%" | <math>\log_9{\displaystyle \frac{1}{3}}</math>
|c)
|c)
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|width="33%" | <math>\log_2{0{,}125}</math>
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|width="33%" | <math>\log_2{0\textrm{.}125}</math>
|-
|-
|d)
|d)
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|width="33%" | <math>\log_a{\bigl(a^2\sqrt{a}\,\bigr)}</math>
|width="33%" | <math>\log_a{\bigl(a^2\sqrt{a}\,\bigr)}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:3|Lösning a|Lösning 3.3:3a|Lösning b|Lösning 3.3:3b|Lösning c|Lösning 3.3:3c|Lösning d|Lösning 3.3:3d|Lösning e|Lösning 3.3:3e|Lösning f|Lösning 3.3:3f|Lösning g|Lösning 3.3:3g|Lösning h|Lösning 3.3:3h}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:3|Solution a|Solution 3.3:3a|Solution b|Solution 3.3:3b|Solution c|Solution 3.3:3c|Solution d|Solution 3.3:3d|Solution e|Solution 3.3:3e|Solution f|Solution 3.3:3f|Solution g|Solution 3.3:3g|Solution h|Solution 3.3:3h}}
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===Övning 3.3:4===
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===Exercise 3.3:4===
<div class="ovning">
<div class="ovning">
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F&ouml;renkla
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Simplify
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%" | <math>\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}</math>
|width="33%" | <math>\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:4|Lösning a|Lösning 3.3:4a|Lösning b|Lösning 3.3:4b|Lösning c|Lösning 3.3:4c}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:4|Solution a|Solution 3.3:4a|Solution b|Solution 3.3:4b|Solution c|Solution 3.3:4c}}
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===Övning 3.3:5===
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===Exercise 3.3:5===
<div class="ovning">
<div class="ovning">
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F&ouml;renkla
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Simplify
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%" | <math>\left(e^{\ln{e}}\right)^2</math>
|width="33%" | <math>\left(e^{\ln{e}}\right)^2</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:5|Lösning a|Lösning 3.3:5a|Lösning b|Lösning 3.3:5b|Lösning c|Lösning 3.3:5c|Lösning d|Lösning 3.3:5d|Lösning e|Lösning 3.3:5e|Lösning f|Lösning 3.3:5f}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:5|Solution a|Solution 3.3:5a|Solution b|Solution 3.3:5b|Solution c|Solution 3.3:5c|Solution d|Solution 3.3:5d|Solution e|Solution 3.3:5e|Solution f|Solution 3.3:5f}}
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===Övning 3.3:6===
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===Exercise 3.3:6===
<div class="ovning">
<div class="ovning">
{| width="100%"
{| width="100%"
| width="100%" |
| width="100%" |
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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''):
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Use the calculator on the right to calculate the following to three decimal places. The button <tt>LN</tt> signifies the natural logarithm with base ''e''.
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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||{{LOGCALCULATOR}}
||{{LOGCALCULATOR}}
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 3.3:6|Lösning a|Lösning 3.3:6a|Lösning b|Lösning 3.3:6b|Lösning c|Lösning 3.3:6c}}
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</div>{{#NAVCONTENT:Answer|Answer 3.3:6|Solution a|Solution 3.3:6a|Solution b|Solution 3.3:6b|Solution c|Solution 3.3:6c}}

Current revision

       Theory          Exercises      


Exercise 3.3:1

Solve the following equations for \displaystyle x.

a) \displaystyle 10^x=1\,000 b) \displaystyle 10^x=0\textrm{.}1
c) \displaystyle \displaystyle \frac{1}{10^x}=100 d) \displaystyle \displaystyle \frac{1}{10^x}=0\textrm{.}000\,1
Answer
Answer 3.3:1
Solution a
Solution 3.3:1a
Solution b
Solution 3.3:1b
Solution c
Solution 3.3:1c
Solution d
Solution 3.3:1d

Exercise 3.3:2

Calculate

a) \displaystyle \lg{ 0\textrm{.}1} b) \displaystyle \lg{ 10\,000} c) \displaystyle \lg {0\textrm{.}001} d) \displaystyle \lg {1}
e) \displaystyle 10^{\lg{2}} f) \displaystyle \lg{10^3} g) \displaystyle 10^{-\lg{0\textrm{.}1}} h) \displaystyle \lg{\displaystyle \frac{1}{10^2}}
Answer
Answer 3.3:2
Solution a
Solution 3.3:2a
Solution b
Solution 3.3:2b
Solution c
Solution 3.3:2c
Solution d
Solution 3.3:2d
Solution e
Solution 3.3:2e
Solution f
Solution 3.3:2f
Solution g
Solution 3.3:2g
Solution h
Solution 3.3:2h

Exercise 3.3:3

Calculate

a) \displaystyle \log_2{8} b) \displaystyle \log_9{\displaystyle \frac{1}{3}} c) \displaystyle \log_2{0\textrm{.}125}
d) \displaystyle \log_3{\left(9\cdot3^{1/3}\right)} e) \displaystyle 2^{\log_{\scriptstyle2}{4}} f) \displaystyle \log_2{4}+\log_2{\displaystyle \frac{1}{16}}
g) \displaystyle \log_3{12}-\log_3{4} h) \displaystyle \log_a{\bigl(a^2\sqrt{a}\,\bigr)}
Answer
Answer 3.3:3
Solution a
Solution 3.3:3a
Solution b
Solution 3.3:3b
Solution c
Solution 3.3:3c
Solution d
Solution 3.3:3d
Solution e
Solution 3.3:3e
Solution f
Solution 3.3:3f
Solution g
Solution 3.3:3g
Solution h
Solution 3.3:3h

Exercise 3.3:4

Simplify

a) \displaystyle \lg{50}-\lg{5} b) \displaystyle \lg{23}+\lg{\displaystyle \frac{1}{23}} c) \displaystyle \lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}
Answer
Answer 3.3:4
Solution a
Solution 3.3:4a
Solution b
Solution 3.3:4b
Solution c
Solution 3.3:4c

Exercise 3.3:5

Simplify

a) \displaystyle \ln{e^3}+\ln{e^2} b) \displaystyle \ln{8}-\ln{4}-\ln{2} c) \displaystyle (\ln{1})\cdot e^2
d) \displaystyle \ln{e}-1 e) \displaystyle \ln{\displaystyle \frac{1}{e^2}} f) \displaystyle \left(e^{\ln{e}}\right)^2
Answer
Answer 3.3:5
Solution a
Solution 3.3:5a
Solution b
Solution 3.3:5b
Solution c
Solution 3.3:5c
Solution d
Solution 3.3:5d
Solution e
Solution 3.3:5e
Solution f
Solution 3.3:5f

Exercise 3.3:6

Use the calculator on the right to calculate the following to three decimal places. The button LN signifies the natural logarithm with base e.

a) \displaystyle \log_3{4}
b) \displaystyle \lg{46}
c) \displaystyle \log_3{\log_2{(3^{118})}}





Answer
Answer 3.3:6
Solution a
Solution 3.3:6a
Solution b
Solution 3.3:6b
Solution c
Solution 3.3:6c
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