Because \(f (x) = b^x\) and \(g(x) = \log_b x\) are inverse functions for \(b\gt 0, ~b\ne 1\text{,}\)
\begin{equation*}
\log_b b^x = x~~\text{, for all }x~~\text{ and }~~~b^{\log_b x} = x~~\text{, for }x\gt 0
\end{equation*}
Section 10.4 Chapter 10 Summary and Review
Subsection 10.4.1 Glossary
- inverse function
- logarithmic function
- logarithmic equation
- compound interest
- natural base
- natural log function
- natural exponential function
- continuous compounding
- log scale
- pH value
- decibels
- Richter magnitude
Subsection 10.4.2 Key Concepts
- Two functions are called inverse functions if each function undoes the effects of the other.
- We can make a table of values for the inverse function by interchanging the columns of a table for \(f\text{.}\)
- If we apply the inverse function to the output of \(f\text{,}\) we return to the original input value.
- The graphs of \(f\) and its inverse function are symmetric about the line \(y = x\).
- We define the logarithmic function, \(g(x) = \log_b x\text{,}\) which takes the log base \(b\) of its input values. The log function \(g(x) = \log_b x\) is the inverse of the exponential function \(f(x) = b^x\text{.}\)
- A logarithmic equation is one where the variable appears inside of a logarithm. We can solve logarithmic equations by converting to exponential form.
Steps for Solving Logarithmic Equations.
- Use the properties of logarithms to combine all logs into one log.
- Isolate the log on one side of the equation.
- Convert the equation to exponential form.
- Solve for the variable.
- Check for extraneous solutions.
- The natural base is an irrational number called \(e\text{,}\) where\begin{equation*} e\approx 2.71828182845 \end{equation*}
- The natural exponential function is the function \(f(x) = e^x\text{.}\) The natural log function is the function \(g(x) = \ln x = \log_e x\text{.}\)
Conversion Formulas for the Natural Base.
\begin{equation*} y = \ln x~~~~~~\text{if and only if}~~~~~~e^y = x \end{equation*}Properties of Natural Logarithms.
If \(x, y \gt 0\text{,}\) then- \(\displaystyle \ln{(xy)} = \ln{x} + \ln{y}\)
- \(\displaystyle \ln\dfrac{x}{y} = \ln x - \ln y\)
- \(\displaystyle \ln{x^k} = k \ln x \)
also\begin{equation*} \ln e^x = x~~~~\text{for all }x\text{ and}~~~~e^{\ln x}=x\text{, for }x \gt 0 \end{equation*}- We use the natural logarithm to solve exponential equations with base \(e\text{.}\)
Exponential Growth and Decay.
The function\begin{equation*} P(t) = P_0 e^{kt} \end{equation*}describes exponential growth if \(k \gt 0\text{,}\) and exponential decay if \(k \lt 0\text{.}\)- Continuous compounding: The amount accumulated in an account after \(t\) years at interest rate \(r\) compounded continuously is given by\begin{equation*} A(t) = Pe^{rt} \end{equation*}where \(P\) is the principal invested.
- A log scale is useful for plotting values that vary greatly in magnitude. We plot the log of the variable, instead of the variable itself.
- A log scale is a multiplicative scale: Each increment of equal length on the scale indicates that the value is multiplied by an equal amount.
- The pH value of a substance is defined by the formula\begin{equation*} \text{pH}=-\log_{10}[H^+] \end{equation*}where \([H^+]\) denotes the concentration of hydrogen ions in the substance.
- The loudness of a sound is measured in decibels, \(D\text{,}\) by\begin{equation*} D=10 \log_{10}\left(\frac{I}{10^{-12}}\right) \end{equation*}where \(I\) is the intensity of its sound waves (in watts per square meter).
- The Richter magnitude, \(M\text{,}\) of an earthquake is given by\begin{equation*} M=\log_{10}\left(\frac{A}{A_0} \right) \end{equation*}where \(A\) is the amplitude of its seismographic trace and \(A_0\) is the amplitude of the smallest detectable earthquake.
- A difference of \(K\) units on a logarithmic scale corresponds to a factor of \(10^K\) units in the value of the variable.
Exercises 10.4.3 Chapter 10 Review Problems
Exercise Group.
For Problems 1 and 2, make a table of values for the function and sketch a graph.
1.
\(f(x)=\log_4 {x}\)
2.
\(f(x)=\log_{1/4} {x}\)
Exercise Group.
For Problems 3 and 4, simplify.
3.
- \(\displaystyle 10^{\log_6 n}\)
- \(\displaystyle \log_2{4^{x+3}}\)
4.
- \(\displaystyle \log {100^x}\)
- \(\displaystyle 3^{2 \log_3 t}\)
Exercise Group.
For Problems 5-12, solve.
5.
\(\log_{3}\dfrac{1}{3}=y \)
Answer.
\(-1\)
6.
\(\log_{3}x=4 \)
7.
\(\log_{2}y=-1 \)
Answer.
\(\dfrac{1}{2} \)
8.
\(\log_{5}y=-2 \)
9.
\(\log_{b} 16=2 \)
Answer.
\(4 \)
10.
\(\log_{b}9=\dfrac{1}{2} \)
11.
\(\log_{4}\left(\dfrac{1}{2}t+1\right)=-2 \)
Answer.
\(\dfrac{-15}{8} \)
12.
\(\log_{2}(3x-1)=3 \)
Exercise Group.
For Problems 13-16, solve.
13.
\(\log_3 x + \log_3 4 = 2\)
Answer.
\(\dfrac{9}{4} \)
14.
\(\log_2(x + 2) - \log_2 3 = 6\)
15.
\(\log_{10}(x-1) + \log_{10} (x+2) = 1\)
Answer.
\(3 \)
16.
\(\log_{10}(x + 2) - \log_{10} (x-3) = 1\)
Exercise Group.
For Problems 17-22, solve.
17.
\(e^x=4.7 \)
Answer.
\(x\approx 1.548 \)
18.
\(e^x=0.5 \)
19.
\(\ln x =6.02 \)
Answer.
\(x\approx 411.58 \)
20.
\(\ln x=-1.4 \)
21.
\(4.73=1.2e^{0.6x} \)
Answer.
\(x\approx 2.286 \)
22.
\(1.75=0.3e^{-1.2x} \)
Exercise Group.
For Problems 23-26, simplify.
23.
\(e^{(\ln x)/2} \)
Answer.
\(\sqrt{x} \)
24.
\(\ln \left(\dfrac{1}{e} \right)^{2n} \)
25.
\(\ln \left(\dfrac{e^k}{e^3} \right) \)
Answer.
\(k-3 \)
26.
\(e^{\ln(e+x)} \)
27.
In 1970, the population of New York City was 7,894,862. In 1980, the population had fallen to 7,071,639.
- Write an exponential function using base \(e\) for the population of New York over that decade.
- By what percent did the population decline annually?
Answer.
- \(\displaystyle P = 7,894,862e^{-0.011t}\)
- 1.095%
28.
In 1990, the population of New York City was 7,322,564. In 2000, the population was 8,008,278.
- Write an exponential function using base \(e\) for the population of New York over that decade.
- By what percent did the population increase annually?
29.
You deposit $1000 in a savings account paying 5% interest compounded continuously.
- Find the amount in the account after 7 years.
- How long will it take for the original principal to double?
- Find a formula for the time \(t\) required for the amount to reach \(A\text{.}\)
Answer.
- $1419.07
- 13.9 years
- \(\displaystyle t = 20 \ln\left(\dfrac{A}{1000} \right)\)
30.
The voltage, \(V\text{,}\) across a capacitor in a certain circuit is given by the function
\begin{equation*}
V(t) = 100(1-e^{-0.5t})
\end{equation*}
where \(t\) is the time in seconds.
- Make a table of values and graph \(V(t)\) for \(t = 0\) to \(t = 10\text{.}\)
- Describe the graph. What happens to the voltage in the long run?
- How much time must elapse (to the nearest hundredth of a second) for the voltage to reach 75 volts?
31.
Solve for \(t\text{:}\) \(~~y = 12 e^{-kt} + 6\)
Answer.
\(t=\dfrac{-1}{k}\ln\left(\dfrac{y-6}{12} \right) \)
32.
Solve for \(k\text{:}\) \(~~N = N_0 + 4 \ln(k + 10)\)
33.
Solve for \(M\text{:}\) \(~~Q=\dfrac{1}{t}\left(\dfrac{\log M}{\log N} \right) \)
Answer.
\(M=N^{Qt} \)
34.
Solve for \(t\text{:}\) \(~~C_H = C_L\cdot 10^{k}t \)
35.
Express \(P(t) = 750e^{0.32t}\) in the form \(P(t) = P_0b^t\text{.}\)
Answer.
\(P (t) = 750 (1.3771)^t\)
36.
Express \(P(t) = 80e^{-0.6t}\) in the form \(P(t) = P_0 b^t\text{.}\)
37.
Express \(N(t)=600(0.4)^{t}\) in the form \(N(t) = N_0 e^{kt}\text{.}\)
Answer.
\(N(t) = 600 e^{-0.9163t}\)
38.
Express \(N(t) =100(1.06)^{t}\) in the form \(N(t) = N_0 e^{kt}\text{.}\)
39.
Plot the values on a log scale.
| \(x\) | \(0.04\) | \(45\) | \(1200\) | \(560,000\) |
Answer.
40.
Plot the values on a log scale.
| \(x\) | \(0.0007\) | \(0.8\) | \(3.2\) | \(2500\) |
41.
The graph describes a network of streams near Santa Fe, New Mexico. It shows the number of streams of a given order, which is a measure of their size. Use the graph to estimate the number of streams of orders 3, 4, 8, and 9. (Source: Leopold, Wolman, and Miller)
Answer.
Order 3: 17,000; Order 4: 5000; Order 8: 40; Order 9: 11
42.
Large animals use oxygen more efficiently when running than small animals do. The graph shows the amount of oxygen various animals use, per gram of their body weight, to run 1 kilometer. Estimate the body mass and oxygen use for a kangaroo rat, a dog, and a horse. (Source: Schmidt-Neilsen, 1972)
43.
The loudest sound created in a laboratory registered at 210 decibels. The energy from such a sound is sufficient to bore holes in solid material. Find the intensity of a 210-decibel sound.
44.
The most powerful earthquake ever recorded occurred in Chile on 22 May 1960. The magnitude of the earthquake was approximately 9.5. What was the amplitude of its seismographic trace.
45.
In 2004, a magnitude 9.0 earthquake struck Sumatra in Indonesia. How much more powerful was this quake than the 1906 San Francisco earthquake of magnitude 8.3?
46.
The sound of rainfall registers at 50 decibels. What is the decibel level of a sound twice as loud?
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