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# 4.3E: Exercises

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## 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 $$b>0, b\neq 1$$

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?

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?

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

$$a^c=b$$

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

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

$$x^y=64$$

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

Exercise $$\PageIndex{11}$$

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

$$15^b=a$$

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

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

$$13^a=142$$

14) $$\log(v)=t$$

15) $$\ln(w)=n$$

$$e^n=w$$

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

16) $$4^x=y$$

17) $$c^d=k$$

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

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

Exercise $$\PageIndex{19}$$

$$19^x=y$$

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

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

21) $$n^4 = 103$$

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

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

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

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

24) $$10^a=b$$

25) $$e^k=h$$

$$\ln(w)=n$$

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

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

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

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

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

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

$$x = 3^3 = 27$$

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

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

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

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

Exercise $$\PageIndex{33}$$

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

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

34) $$\log (x)=3$$

35) $$\ln(x)=2$$

$$x=e^2$$

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

36) $$\log (100^8)$$

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

$$32$$

38) $$2\log (.0001)$$

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

$$1.06$$

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

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

$$14.125$$

### Numeric

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

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

Exercise $$\PageIndex{43}$$

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

$$\dfrac{1}{2}$$

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

45) $$6\log _8(4)$$

$$4$$

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

46) $$\log (10,000)$$

47) $$\log (0.001)$$

$$-3$$

48) $$\log (1)+7$$

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

$$-12$$

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

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

51) $$\ln (1)$$

$$0$$

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

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

$$10$$

### Technology

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

54) $$\log (0.04)$$

55) $$\ln (15)$$

$$2.708$$

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

Exercise $$\PageIndex{57}$$

$$\log (\sqrt{2})$$

$$0.151$$

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

### Extensions

59) Is $$x=0$$

No, the function has no defined value for $$x=0$$$.$ 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 $$f(x)=\log (x)$$

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

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

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

63) Is the following true:

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

$$2$$
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.