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Exercises 7.7 Exercises
1. Graphs of Exponential and Logarithmic Functions.
Sketch the graph of \(y = 2^x\) and label at least three points inclusing the \(y\) -intercept. State the domain and range of the graph and identfy the horizontal asymptote. Then plot its inverse function’s graph \(y = \log_2(x)\text{.}\) Label the corresponding points on the new curve, state the domain and range, and identify the vertical asymptote.
2. Graphs of Exponential and Logarithmic Functions.
Sketch the graph of each equation below.
\(\displaystyle y = 2^{-x}\)
\(\displaystyle y = -2^{x}\)
\(\displaystyle y = 2^{x}-1\)
\(\displaystyle y = 2^{x-1}\)
\(\displaystyle y = \log_2(x+1)\)
\(\displaystyle y = \log_2(x)+1\)
Hint .
\(y = 2^{-x}\) is the graph of \(y=2^x\) reflected across the \(y\) -axis.
\(y = -2^{x}\) is the graph of \(y=2^x\) reflected across the \(x\) -axis.
\(y = 2^{x}-1\) is the graph of \(y=2^x\) translated down one unit.
\(y = 2^{x-1}\) is the graph of \(y=2^x\) translated right one unit.
\(y = \log_2(x-1)\) is the graph of \(y=\log_2(x)\) translated left one unit.
\(y = \log_2(x)-1\) is the graph of \(y=\log_2(x)\) translated up one unit.
3. Evaluating Logarithmic Functions.
Evaluate each logarithm without using a calculator.
\(\displaystyle \log_3(9)\)
\(\displaystyle \log_3\left(\frac{1}{3}\right)\)
\(\displaystyle \log_{\frac{1}{3}}\left(\frac{1}{3}\right)\)
\(\displaystyle \log_{\frac{1}{3}}\left(3\right)\)
\(\displaystyle \log_{3}\left(\frac{1}{\sqrt[3]{3}}\right)\)
\(\displaystyle \log_{3}\left(1\right)\)
Answer .
\(\displaystyle \log_3(9) = 2\)
\(\displaystyle \log_3\left(\frac{1}{3}\right) = -1\)
\(\displaystyle \log_{\frac{1}{3}}\left(\frac{1}{3}\right) = 1\)
\(\displaystyle \log_{\frac{1}{3}}\left(3\right) = -1\)
\(\displaystyle \log_{3}\left(\frac{1}{\sqrt[3]{3}}\right)= - 1/3\)
\(\displaystyle \log_{3}\left(1\right) = 0\)
4. Evaluating Logarithmic Functions.
Evaluate each natural logarithm without using a calculator.
\(\displaystyle \ln\left(\sqrt{e}\right)\)
\(\displaystyle \ln\left(\frac{1}{e^2}\right)\)
\(\displaystyle \ln\left(\left|-e\right|\right)\)
\(\displaystyle \ln\left(1\right)\)
Answer .
\(\displaystyle \ln\left(\sqrt{e}\right) = 1/2\)
\(\displaystyle \ln\left(\frac{1}{e^2}\right) = -2\)
\(\displaystyle \ln\left(\left|-e\right|\right) = 1\)
\(\displaystyle \ln\left(1\right) = 0\)
5. Converting to Logarithmic Form.
Rewrite each exponential equation in logarithmic form, but do not attempt to solve.
\(\displaystyle 5^{x-1} = 6\)
\(\displaystyle a^5 = x\)
\(\displaystyle e^{x^2} = x\)
\(\displaystyle \frac{1}{e^x} = \pi\)
Answer .
\(\displaystyle 5^{x-1} = 6 \quad \Rightarrow \quad \boxed{\log_5(6)=x-1}\)
\(\displaystyle a^5 = x \quad \Rightarrow \quad \boxed{\log_a(x)=5}\)
\(\displaystyle e^{x^2} = x \quad \Rightarrow \quad \boxed{\ln(x) = x^2}\)
\(\displaystyle \frac{1}{e^x} = \pi \quad \Rightarrow \quad \boxed{\ln(\pi) = - x}\)
6. Converting to Exponential Form.
Rewrite each logarithmic equation in exponential form, but do not attempt to solve.
\(\displaystyle \log_4 (x+y) = y\)
\(\displaystyle \log_{0.25}(\pi) = x\)
\(\displaystyle \log_b(x) = z\)
\(\displaystyle \ln(t)=0.5\)
Answer .
\(\displaystyle \log_4 (x+y) = y \quad \Rightarrow \quad \boxed{4^y = x+y}\)
\(\displaystyle \log_{0.25}(\pi) = x \quad \Rightarrow \quad \boxed{(0.25)^x = \pi}\)
\(\displaystyle \log_b(x) = z \quad \Rightarrow \quad \boxed{b^z = x}\)
\(\displaystyle \ln(t)=0.5 \quad \Rightarrow \quad \boxed{e^{0.5} = t}\)
7. Properties of Logarithms.
Given that \(\ln(x) = 8\text{,}\) \(\ln(y)= -1\text{,}\) and \(\ln(z)=3\text{,}\) use the properties of logarithms to find the value of the expresion.
\(\displaystyle \ln\left(x y z^2\right)\)
\(\displaystyle \ln\left(\frac{x y}{z^2}\right)\)
\(\displaystyle \ln\left(\frac{x}{yz^2}\right)\)
\(\displaystyle \ln\left(\sqrt[3]{x}\right)\)
Answer .
\(\displaystyle \ln\left(x y z^2\right) = 8-1+6 = 13\)
\(\displaystyle \ln\left(\frac{x y}{z^2}\right) = 8-1-6 = 1\)
\(\displaystyle \ln\left(\frac{x}{yz^2}\right) = 8+1-6 = 3\)
\(\displaystyle \ln\left(\sqrt[3]{x}\right) = \frac{8}{3}\)
8. Properties of Logarithms.
Express as a single logarithm.
\(\displaystyle \ln(10)-\ln(5)+\frac{1}{2}\ln(9)\)
\(\displaystyle \ln(a)+\ln(b)-2\ln(c)\)
\(\displaystyle 2\ln(x+1)-\ln(x-1)-4\ln(x)\)
Answer .
\(\displaystyle \ln(10)-\ln(5)+\frac{1}{2}\ln(9) = \ln(6)\)
\(\displaystyle \ln(a)+\ln(b)-2\ln(c) = \ln\left(\frac{ab}{c^2}\right) \)
\(\displaystyle 2\ln(x+1)-\ln(x-1)-4\ln(x) = \ln\left(\frac{(x+1)^2}{(x-1)x^4}\right)\)
9. Properties of Logarithms.
Simplify each expression as much as possible.
\(\displaystyle 2^{\log_2(5)}\)
\(\displaystyle 2^{2\log_2(5)}\)
\(\displaystyle 8^{\log_2(5)}\)
\(\displaystyle 2^{2+\log_2(5)}\)
\(\displaystyle e^{-\ln(5)}\)
\(\displaystyle e^{2\ln(5)}\)
Answer .
\(\displaystyle 2^{\log_2(5)} = 5\)
\(\displaystyle 2^{2\log_2(5)} = 25\)
\(\displaystyle 8^{\log_2(5)} = 125\)
\(\displaystyle 2^{2+\log_2(5)} = 4\cdot 5 = 20\)
\(\displaystyle e^{-\ln(5)} = \frac{1}{5}\)
\(\displaystyle e^{2\ln(5)} = 25\)
10. Solving Exponential Equations.
Solve each equation. Express your answer in terms of logarithms and use a calculator to obtain an approximation of the solution(s).
\(\displaystyle 2^{t} = 10\)
\(\displaystyle 2^{t}-2 = 10\)
\(\displaystyle 2^{t-2} = 10\)
\(\displaystyle 2e^{t} = 10\)
\(\displaystyle e^{-2t} = 10\)
\(\displaystyle e^{t-2} = 10\)
Answer .
\(\displaystyle 2^{t} = 10 \quad \Rightarrow \quad t = \log_2(10) = \frac{\ln(10)}{\ln(2)} \approx 3.32193\)
\(\displaystyle 2^{t}-2 = 10 \quad \Rightarrow \quad t = \frac{\ln(12)}{\ln(2)} \approx 3.58496\)
\(\displaystyle 2^{t-2} = 10 \quad \Rightarrow \quad t = \frac{\ln(10)}{\ln(2)} + 2 \approx 5.32193\)
\(\displaystyle 2e^{t} = 10 \quad \Rightarrow \quad t = \ln(5) \approx 1.60944\)
\(\displaystyle e^{-2t} = 10 \quad \Rightarrow \quad t = -\frac{\ln(10)}{2} \approx -1.15129\)
\(\displaystyle e^{t-2} = 10 \quad \Rightarrow \quad t = \ln(10)+2 \approx 4.30259\)
11. Solving Exponential Equations.
The voltage across a capicitor after \(t\) milliseconds is
\begin{equation*}
V(t) = 10 e^{-0.05 t} + 2.
\end{equation*}
Find the initial voltage.
Describe the voltage after a long period of time.
Find how long it takes for the voltage to be 5 volts.
Answer .
\(V(0)=10+2 = 12\) volts.
The voltage will approach 2 volts.
Solve \(V(t) = 5\) to find that \(t \approx 24.1\) ms.
12. Avoiding Common Mistakes.
Is \(\ln(A+B)\) the same as \(\ln(A)+\ln(B)\text{?}\) Is \(\left(\ln(A)\right)^n\) the same as \(n \ln(A)\text{?}\) Is \(\ln(A/B)\) the same as \(\frac{\ln(A)}{\ln(B)}\text{?}\)
13. Properties of Logarithms.
Prove that \(\ln(A/B) = \ln(A)-\ln(B)\) and that \(\ln(A^n) = n \ln(A)\text{.}\)
Hint . Let \(x=\ln(A)\) and \(y = \ln(B)\text{.}\) Then rewrite each equation in exponential form.
14. Properties of Logarithms.
Prove the change of base formula for logarithms:
\begin{equation*}
\log_a(y) = \frac{\ln(y)}{\ln(a)},
\end{equation*}
where \(a \gt 0, a\neq 1\) and \(y \gt 0\text{.}\)
Hint . Solve the exponential equation \(a^x = y\) for \(x\) two different ways. Why must the solutions agree?
