4.3E: Exercises

4.3: Logarithmic Functions

The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

Verbal

1) What is a base $b$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $b^y=x$ and $\log _bx=y$ for $b>0, b\neq 1$.

Answer

A logarithm is an exponent. Specifically, it is the exponent to which a base $b$ is raised to produce a given value. In the expressions given, the base $b$ has the same value. The exponent, $y$$,$in the expression $b^y$ can also be written as the logarithm, $\log _bx=y$$,$and the value of $x$ is the result of raising $b$ to the power of $y$.

2) How is the logarithmic function $f(x)=\log _bx$ related to the exponential function $g(x)=b^x$? What is the result of composing these two functions?

3) How can the logarithmic equation $\log _bx=y$ be solved for $x$ using the properties of exponents?

Answer

Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation $b^y = x$$,$ and then properties of exponents can be applied to solve for $x$$.$

4) Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $b$$,$ and how does the notation differ?

5) Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $b$$,$ and how does the notation differ?

Answer

The natural logarithm is a special case of the logarithm with base $b$ in that the natural log always has base $e$.Rather than notating the natural logarithm as $\log_{e}(x)$$,$the notation used is $\ln (x)$$.$

Algebraic

For the following exercises, rewrite each equation in exponential form.

6) $\log_{4}(q)=m$

7) $\log_{a}(b)=c$

Answer

$a^c=b$

8) $\log_{16}(y)=x$

9) $\log_{x}(64)=y$

Answer

$x^y=64$

10) $\log_{y}(x)=-11$

Exercise $\PageIndex{11}$

$\log_{15}(a)=b$

Answer

$15^b=a$

12) $\log_{y}(137)=x$

13) $\log_{13}(142)=a$

Answer

$13^a=142$

14) $\log(v)=t$

15) $\ln(w)=n$

Answer

$e^n=w$

For the following exercises, rewrite each equation in logarithmic form.

16) $4^x=y$

17) $c^d=k$

Answer

$\log_{c}(k)=d$

18) $m^{-7}=n$

Exercise $\PageIndex{19}$

$19^x=y$

Answer

$\log_{19}(y)=x$

20) $x^{-\frac{10}{13}}=y$

21) $n^4 = 103$

Answer

$\log_{n}(103)=4$

22) $\left ( \dfrac{7}{5} \right )^m=n$

23) $y^x=\dfrac{39}{100}$

Answer

$\log_{y}\left ( \dfrac{39}{100} \right )=x$

24) $10^a=b$

25) $e^k=h$

Answer

$\ln(w)=n$

For the following exercises, solve for $x$ by converting the logarithmic equation to exponential form.

26) $\log_{3}(x)=2$

27) $\log_{2}(x)=-3$

Answer

$x=2^{-3}=\dfrac{1}{8}$

28) $\log_{5}(x)=2$

29) $\log_{3}(x)=3$

Answer

$x = 3^3 = 27$

30) $\log_{2}(x)=6$

31) $\log_{9}(x)=\dfrac{1}{2}$

Answer

$x=9^{\frac{1}{2}}=3$

32) $\log_{18}(x)=2$

Exercise $\PageIndex{33}$

$\log_{6}(x)=-3$

Answer

$x=6^{-3}=\dfrac{1}{216}$

34) $\log (x)=3$

35) $\ln(x)=2$

Answer

$x=e^2$

For the following exercises, use the definition of common and natural logarithms to simplify.

36) $\log (100^8)$

37) $10^{\log (32)}$

Answer

$32$

38) $2\log (.0001)$

39) $e^{\ln (1.06)}$

Answer

$1.06$

40) $\ln (e^{-5.03})$

41) $e^{\ln (10.125)}+4$

Answer

$14.125$

Numeric

For the following exercises, evaluate the base $b$ logarithmic expression without using a calculator.

42) $\log _3\left ( \frac{1}{27} \right )$

Exercise $\PageIndex{43}$

$\log _6(\sqrt{6})$

Answer

$\dfrac{1}{2}$

44) $\log _2\left ( \frac{1}{8} \right )+4$

45) $6\log _8(4)$

Answer

$4$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

46) $\log (10,000)$

47) $\log (0.001)$

Answer

$-3$

48) $\log (1)+7$

49) $2\log (100^{-3})$

Answer

$-12$

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

50) $\ln \left ( e^{\frac{1}{3}} \right )$

51) $\ln (1)$

Answer

$0$

52) $\ln \left ( e^{-0.225} \right )-3$

53) $25\ln \left ( e^{\frac{2}{5}} \right )$

Answer

$10$

Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

54) $\log (0.04)$

55) $\ln (15)$

Answer

$2.708$

56) $\ln \left ( {\frac{4}{5}} \right )$

Exercise $\PageIndex{57}$

$\log (\sqrt{2})$

Answer

$0.151$

58) $\ln (\sqrt{2})$

Extensions

59) Is $x=0$ in the domain of the function $f(x)=\log x$? If so, what is the value of the function when $x=0$? Verify the result.

Answer

No, the function has no defined value for $x=0$$\mathrm{.}$ To verify, suppose $x=0$ is in the domain of the function $f(x)=\log (x)$$.$ Then there is some number $n$ such that $n=\log(0)$$.$ Rewriting as an exponential equation gives: $10^n=0$$,$ which is impossible since no such real number $n$ exists. Therefore, $x=0$ is not the domain of the function $f(x)=\log (x)$.

60) Is $f(x)=0$ in the range of the function $f(x)=\log (x)$?If so, for what value of $x$?Verify the result.

61) Is there a number $x$ such that $\ln x = 2$? If so, what is that number? Verify the result.

Answer

Yes. Suppose there exists a real number $x$ such that $\ln x = 2$$\mathrm{.}$Rewriting as an exponential equation gives $x=e^2$$,$ which is a real number. To verify, let $x=e^2$$.$Then, by definition, $\ln (x)=\ln \left ( e^2 \right ) = 2$$\mathrm{.}$

62) Is the following true: $\frac{\log _3(27)}{\log _4\left ( \frac{1}{64} \right )}=-1$ $\frac{{\log }_3(27)}{{\log }_4\left(\frac{1}{64}\right)}=\mathrm{-1}?$Verify the result.

63) Is the following true: $\frac{\ln (e^{1.725})}{\ln (1)}=1.725$ Verify the result.

Answer

No; $\ln (1) =0$, so $\frac{\ln (e^{1.725})}{\ln (1)}=1.725$ is undefined.

Real-World Applications

64) The exposure index $EI$ for a $35$ millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation $EI=\log _2\left ( \frac{f^2}{t} \right )$$,$ where $f$ is the “f-stop” setting on the camera, and $t$ is the exposure time in seconds. Suppose the f-stop setting is $8$ and the desired exposure time is $2$seconds. What will the resulting exposure index be?

65) Refer to the previous exercise. Suppose the light meter on a camera indicates an $EI$ of  $-2$$,$ and the desired exposure time is $16$ seconds. What should the f-stop setting be?

Answer

$2$

66) The intensity levels $I$ of two earthquakes measured on a seismograph can be compared by the formula $\log \left ( \frac{I_1}{I_2} \right )=M_1-M_2$where $M$ is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude $6.1$ hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of $9.0$. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.