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 \(b>0, b\neq 1\)
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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?
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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?
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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) \(\log_{a}(b)=c\)
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\(a^c=b\)
8) \(\log_{16}(y)=x\)
9) \(\log_{x}(64)=y\)
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\(x^y=64\)
10) \(\log_{y}(x)=-11\)
Exercise \(\PageIndex{11}\)
\(\log_{15}(a)=b\)
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\(15^b=a\)
12) \(\log_{y}(137)=x\)
13) \(\log_{13}(142)=a\)
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\(13^a=142\)
14) \(\log(v)=t\)
15) \(\ln(w)=n\)
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\(e^n=w\)
For the following exercises, rewrite each equation in logarithmic form.
16) \(4^x=y\)
17) \(c^d=k\)
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\(\log_{c}(k)=d\)
18) \(m^{-7}=n\)
Exercise \(\PageIndex{19}\)
\(19^x=y\)
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\(\log_{19}(y)=x\)
20) \(x^{-\frac{10}{13}}=y\)
21) \(n^4 = 103\)
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\(\log_{n}(103)=4\)
22) \(\left ( \dfrac{7}{5} \right )^m=n\)
23) \(y^x=\dfrac{39}{100}\)
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\(\log_{y}\left ( \dfrac{39}{100} \right )=x\)
24) \(10^a=b\)
25) \(e^k=h\)
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\(\ln(w)=n\)
For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form.
26) \(\log_{3}(x)=2\)
27) \(\log_{2}(x)=-3\)
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\(x=2^{-3}=\dfrac{1}{8}\)
28) \(\log_{5}(x)=2\)
29) \(\log_{3}(x)=3\)
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\(x = 3^3 = 27\)
30) \(\log_{2}(x)=6\)
31) \(\log_{9}(x)=\dfrac{1}{2}\)
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\(x=9^{\frac{1}{2}}=3\)
32) \(\log_{18}(x)=2\)
Exercise \(\PageIndex{33}\)
\(\log_{6}(x)=-3\)
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\(x=6^{-3}=\dfrac{1}{216}\)
34) \(\log (x)=3\)
35) \(\ln(x)=2\)
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\(x=e^2\)
For the following exercises, use the definition of common and natural logarithms to simplify.
36) \(\log (100^8)\)
37) \(10^{\log (32)}\)
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\(32\)
38) \(2\log (.0001)\)
39) \(e^{\ln (1.06)}\)
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\(1.06\)
40) \(\ln (e^{-5.03})\)
41) \(e^{\ln (10.125)}+4\)
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\(14.125\)
Numeric
For the following exercises, evaluate the base \(b\) logarithmic expression without using a calculator.
42) \(\log _3\left ( \frac{1}{27} \right )\)
Exercise \(\PageIndex{43}\)
\(\log _6(\sqrt{6})\)
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\(\dfrac{1}{2} \)
44) \(\log _2\left ( \frac{1}{8} \right )+4\)
45) \(6\log _8(4)\)
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\(4\)
For the following exercises, evaluate the common logarithmic expression without using a calculator.
46) \(\log (10,000)\)
47) \(\log (0.001)\)
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\(-3\)
48) \(\log (1)+7\)
49) \(2\log (100^{-3})\)
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\(-12\)
For the following exercises, evaluate the natural logarithmic expression without using a calculator.
50) \(\ln \left ( e^{\frac{1}{3}} \right )\)
51) \(\ln (1)\)
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\(0\)
52) \(\ln \left ( e^{-0.225} \right )-3\)
53) \(25\ln \left ( e^{\frac{2}{5}} \right )\)
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\(10\)
Technology
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.
54) \(\log (0.04)\)
55) \(\ln (15)\)
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\(2.708\)
56) \(\ln \left ( {\frac{4}{5}} \right )\)
Exercise \(\PageIndex{57}\)
\(\log (\sqrt{2})\)
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\(0.151\)
58) \(\ln (\sqrt{2})\)
Extensions
59) Is \(x=0\)
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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.
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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:
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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
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\(2\)
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.