3.3 Exercises
From Förberedande kurs i matematik 1
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{| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | ||
| style="border-bottom:1px solid #000" width="5px" | | | style="border-bottom:1px solid #000" width="5px" | | ||
- | {{ | + | {{Not selected tab|[[3.3 Logarithms|Theory]]}} |
- | {{ | + | {{Selected tab|[[3.3 Exercises|Exercises]]}} |
| style="border-bottom:1px solid #000" width="100%"| | | style="border-bottom:1px solid #000" width="100%"| | ||
|} | |} | ||
- | + | ===Exercise 3.3:1=== | |
- | === | + | |
<div class="ovning"> | <div class="ovning"> | ||
- | + | 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) | ||
- | |width="50%" | <math>10^x=0{ | + | |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) | ||
- | |width="50%" | <math>\displaystyle \frac{1}{10^x}=0{ | + | |width="50%" | <math>\displaystyle \frac{1}{10^x}=0\textrm{.}000\,1</math> |
|} | |} | ||
- | </div>{{#NAVCONTENT: | + | </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}} |
- | === | + | ===Exercise 3.3:2=== |
<div class="ovning"> | <div class="ovning"> | ||
- | + | Calculate | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
- | |width="25%" | <math>\lg{ 0{ | + | |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) | ||
- | |width="25%" | <math>\lg {0{ | + | |width="25%" | <math>\lg {0\textrm{.}001}</math> |
|d) | |d) | ||
|width="25%" | <math>\lg {1}</math> | |width="25%" | <math>\lg {1}</math> | ||
Line 43: | Line 42: | ||
|width="25%" | <math>\lg{10^3}</math> | |width="25%" | <math>\lg{10^3}</math> | ||
|g) | |g) | ||
- | |width="25%" | <math>10^{-\lg{0{ | + | |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> | ||
|} | |} | ||
- | </div>{{#NAVCONTENT: | + | </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}} |
- | {{LOGCALCULATOR}} | + | ===Exercise 3.3:3=== |
+ | <div class="ovning"> | ||
+ | Calculate | ||
+ | {| width="100%" cellspacing="10px" | ||
+ | |a) | ||
+ | |width="33%" | <math>\log_2{8}</math> | ||
+ | |b) | ||
+ | |width="33%" | <math>\log_9{\displaystyle \frac{1}{3}}</math> | ||
+ | |c) | ||
+ | |width="33%" | <math>\log_2{0\textrm{.}125}</math> | ||
+ | |- | ||
+ | |d) | ||
+ | |width="33%" | <math>\log_3{\left(9\cdot3^{1/3}\right)}</math> | ||
+ | |e) | ||
+ | |width="33%" | <math>2^{\log_{\scriptstyle2}{4}}</math> | ||
+ | |f) | ||
+ | |width="33%" | <math>\log_2{4}+\log_2{\displaystyle \frac{1}{16}}</math> | ||
+ | |- | ||
+ | |g) | ||
+ | |width="33%" | <math>\log_3{12}-\log_3{4}</math> | ||
+ | |h) | ||
+ | |width="33%" | <math>\log_a{\bigl(a^2\sqrt{a}\,\bigr)}</math> | ||
+ | |} | ||
+ | </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}} | ||
+ | |||
+ | ===Exercise 3.3:4=== | ||
+ | <div class="ovning"> | ||
+ | Simplify | ||
+ | {| width="100%" cellspacing="10px" | ||
+ | |a) | ||
+ | |width="33%" | <math>\lg{50}-\lg{5}</math> | ||
+ | |b) | ||
+ | |width="33%" | <math>\lg{23}+\lg{\displaystyle \frac{1}{23}}</math> | ||
+ | |c) | ||
+ | |width="33%" | <math>\lg{27^{1/3}}+\displaystyle \frac{\lg{3}}{2}+\lg{\displaystyle \frac{1}{9}}</math> | ||
+ | |} | ||
+ | </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}} | ||
+ | |||
+ | ===Exercise 3.3:5=== | ||
+ | <div class="ovning"> | ||
+ | Simplify | ||
+ | {| width="100%" cellspacing="10px" | ||
+ | |a) | ||
+ | |width="33%" | <math>\ln{e^3}+\ln{e^2}</math> | ||
+ | |b) | ||
+ | |width="33%" | <math>\ln{8}-\ln{4}-\ln{2}</math> | ||
+ | |c) | ||
+ | |width="33%" | <math>(\ln{1})\cdot e^2</math> | ||
+ | |- | ||
+ | |d) | ||
+ | |width="33%" | <math>\ln{e}-1</math> | ||
+ | |e) | ||
+ | |width="33%" | <math>\ln{\displaystyle \frac{1}{e^2}}</math> | ||
+ | |f) | ||
+ | |width="33%" | <math>\left(e^{\ln{e}}\right)^2</math> | ||
+ | |} | ||
+ | </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}} | ||
+ | |||
+ | ===Exercise 3.3:6=== | ||
+ | <div class="ovning"> | ||
+ | {| width="100%" | ||
+ | | width="100%" | | ||
+ | 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" | ||
+ | |a) | ||
+ | |width="100%" | <math>\log_3{4}</math> | ||
+ | |- | ||
+ | |b) | ||
+ | |width="100%" | <math>\lg{46}</math> | ||
+ | |- | ||
+ | |c) | ||
+ | |width="100%" | <math>\log_3{\log_2{(3^{118})}}</math> | ||
+ | |} | ||
+ | ||{{LOGCALCULATOR}} | ||
+ | |} | ||
+ | </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
Solution a
Solution b
Solution c
Solution d
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
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
Solution g
Solution h
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
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
Solution g
Solution h
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
Solution a
Solution b
Solution c
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
Solution a
Solution b
Solution c
Solution d
Solution e
Solution f
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.
|
Answer
Solution a
Solution b
Solution c