Logarithms  (Page 3/3)

Solving simple log equations

In grade 10 you solved some exponential equations by trial and error, because you did not know the great power of logarithms yet. Now it is much easier to solve these equations by using logarithms.

For example to solve $x$ in ${25}^x=50$ correct to two decimal places you simply apply the following reasoning. If the LHS = RHS then the logarithm of the LHS must be equal to the logarithm of the RHS. By applying Law 5, you will be able to use your calculator to solve for $x$ .

In general, the exponential equation should be simplified as much as possible. Then the aim is to make the unknown quantity (i.e. $x$ ) the subject of the equation.

For example, the equation

is solved by moving all terms with the unknown to one side of the equation and taking all constants to the other side of the equation

Then, take the logarithm of each side.

Substituting into the original equation, yields

Similarly, $9^{(1-2x)}=3^4$ is solved as follows:

Substituting into the original equation, yields

Exercises

Solve for $x$ :

1. ${log}_{3}x=2$
2. ${10}^{log27}=x$
3. $3^{2x-1}={27}^{2x-1}$

Logarithmic applications in the real world

Logarithms are part of a number of formulae used in the Physical Sciences. There are formulae that deal with earthquakes, with sound, and pH-levels to mention a few. To work out time periods is growth or decay, logs are used to solve the particular equation.

Exercises

1. The population of a certain bacteria is expected to grow exponentially at a rate of 15 % every hour. If the initial population is 5 000, how long will it take for the population to reach 100 000 ?
2. Plus Bank is offering a savings account with an interest rate if 10 % per annum compounded monthly. You can afford to save R 300 per month. How long will it take you to save R 20 000 ? (Give your answer in years and months)

End of chapter exercises

1. Show that
   $${log}_a\left(\frac{x}{y}\right)={log}_a\left(x\right)-{log}_a\left(y\right)$$
2. Show that
   $${log}_a\left(\sqrt[b]{x}\right)=\frac{{log}_a\left(x\right)}{b}$$
3. Without using a calculator show that:
   $$log\frac{75}{16}-2log\frac{5}{9}+log\frac{32}{243}=log2$$
4. Given that $5^n=x$ and $n={log}_{2}y$
   1. Write $y$ in terms of $n$
   2. Express ${log}_{8}4y$ in terms of $n$
   3. Express ${50}^{n+1}$ in terms of $x$ and $y$
5. Simplify, without the use of a calculator:
   1. $8^{\frac{2}{3}}+{log}_{2}32$
   2. ${log}_{3}9-{log}_5\sqrt{5}$
   3. ${\left({\displaystyle \frac{5}{4^{-1}-9^{-1}}}\right)}^{{\displaystyle \frac{1}{2}}}+{log}_3{9}^{2,12}$
6. Simplify to a single number, without use of a calculator:
   1. ${log}_{5}125+{\displaystyle \frac{log32-log8}{log8}}$
   2. $log3-log0,3$
7. Given:     ${log}_{3}6=a$ and ${log}_{6}5=b$
   1. Express ${log}_{3}2$ in terms of $a$ .
   2. Hence, or otherwise, find ${log}_{3}10$ in terms of $a$ and $b$ .
8. Given:     $p{q}^k=q{p}^{-1}$ Prove:     $k=1-2{log}_{q}p$
9. Evaluate without using a calculator: ${\left({log}_{7}49\right)}^5+{log}_5\left({\displaystyle \frac{1}{125}}\right)-13{log}_{9}1$
10. If $log5=0,7$ , determine, without using a calculator :
    1. ${log}_{2}5$
    2. ${10}^{-1,4}$
11. Given:        $M={log}_2(x+3)+{log}_2(x-3)$
    1. Determine the values of $x$ for which $M$ is defined.
    2. Solve for $x$ if $M=4$ .
12. Solve:        ${\left(x^3\right)}^{logx}=10x^2$ (Answer(s) may be left in surd form, if necessary.)
13. Find the value of ${\left({log}_{27}3\right)}^3$ without the use of a calculator.
14. Simplify By using a calculator: ${log}_{4}8+2{log}_3\sqrt{27}$
15. Write $log4500$ in terms of $a$ and $b$ if $2={10}^a$ and $9={10}^b$ .
16. Calculate:        ${\displaystyle \frac{5^{2006}-5^{2004}+24}{5^{2004}+1}}$
17. Solve the following equation for $x$ without the use of a calculator and using the fact that $\sqrt{10}\approx 3,16:$
    $$2log(x+1)={\displaystyle \frac{6}{log(x+1)}}-1$$
18. Solve the following equation for $x$ : $6^{6x}=66$    (Give answer correct to 2 decimal places.)