4.3E: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 by=x and logbx=y for b>0,b≠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, yin the expression by can also be written as the logarithm, logbx=yand the value of x is the result of raising b to the power of y.
2) How is the logarithmic function f(x)=logbx related to the exponential function g(x)=bx? What is the result of composing these two functions?
3) How can the logarithmic equation logbx=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 by=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 loge(x)the notation used is ln(x)
Algebraic
For the following exercises, rewrite each equation in exponential form.
6) log4(q)=m
7) loga(b)=c
- Answer
-
ac=b
8) log16(y)=x
9) logx(64)=y
- Answer
-
xy=64
10) logy(x)=−11
Exercise 4.3E.11
log15(a)=b
- Answer
-
15b=a
12) logy(137)=x
13) log13(142)=a
- Answer
-
13a=142
14) log(v)=t
15) ln(w)=n
- Answer
-
en=w
For the following exercises, rewrite each equation in logarithmic form.
16) 4x=y
17) cd=k
- Answer
-
logc(k)=d
18) m−7=n
Exercise 4.3E.19
19x=y
- Answer
-
log19(y)=x
20) x−1013=y
21) n4=103
- Answer
-
logn(103)=4
22) (75)m=n
23) yx=39100
- Answer
-
logy(39100)=x
24) 10a=b
25) ek=h
- Answer
-
ln(w)=n
For the following exercises, solve for x by converting the logarithmic equation to exponential form.
26) log3(x)=2
27) log2(x)=−3
- Answer
-
x=2−3=18
28) log5(x)=2
29) log3(x)=3
- Answer
-
x=33=27
30) log2(x)=6
31) log9(x)=12
- Answer
-
x=912=3
32) log18(x)=2
Exercise 4.3E.33
log6(x)=−3
- Answer
-
x=6−3=1216
34) log(x)=3
35) ln(x)=2
- Answer
-
x=e2
For the following exercises, use the definition of common and natural logarithms to simplify.
36) log(1008)
37) 10log(32)
- Answer
-
32
38) 2log(.0001)
39) eln(1.06)
- Answer
-
1.06
40) ln(e−5.03)
41) eln(10.125)+4
- Answer
-
14.125
Numeric
For the following exercises, evaluate the base b logarithmic expression without using a calculator.
42) log3(127)
Exercise 4.3E.43
log6(√6)
- Answer
-
12
44) log2(18)+4
45) 6log8(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) 2log(100−3)
- Answer
-
−12
For the following exercises, evaluate the natural logarithmic expression without using a calculator.
50) ln(e13)
51) ln(1)
- Answer
-
0
52) ln(e−0.225)−3
53) 25ln(e25)
- 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(45)
Exercise 4.3E.57
log(√2)
- Answer
-
0.151
58) ln(√2)
Extensions
59) Is x=0
- Answer
-
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: 10n=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 lnx=2? If so, what is that number? Verify the result.
- Answer
-
Yes. Suppose there exists a real number x such that lnx=2Rewriting as an exponential equation gives x=e2 which is a real number. To verify, let x=e2Then, by definition, ln(x)=ln(e2)=2
62) Is the following true: log3(27)log4(164)=−1 Verify the result.
63) Is the following true:
- Answer
-
No; ln(1)=0, so ln(e1.725)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=log2(f2t) 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 2seconds. What will the resulting exposure index be?
65) Refer to the previous exercise. Suppose the light meter on a camera indicates an EI of
- Answer
-
2
66) The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log(I1I2)=M1−M2where 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.