Chapter 10

Graph each of the exponential functions. $$ f(x)=\left(\frac{1}{3}\right)^{x} $$

Solve each logarithmic equation and express irrational solutions in lowest radical form. $$ \ln (3 t-4)-\ln (t+1)=\ln 2 $$

Use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 0.008142\)

Evaluate each logarithmic expression. \(\log _{10} 10\)

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=\frac{1}{x+1} \text { and } g(x)=\frac{1-x}{x} $$

Suppose that a certain radioactive substance has a halflife of 20 years. If there are presently 2500 milligrams of the substance, how much, to the nearest milligram, will remain after 40 years? After 50 years? 625 milligrams; 442 milligrams

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=0.4721 $$

Evaluate each logarithmic expression. \(\log _{10} 0.1\)

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=x \text { and } g(x)=\frac{1}{x} $$

Strontium- 90 has a half-life of 29 years. If there are 400 grams of strontium- 90 initially, how much, to the nearest gram, will remain after 87 years? After 100 years? \(\quad 50\) grams; 37 grams

Graph each of the exponential functions. $$ f(x)=\left(\frac{3}{2}\right)^{x} $$

Solve each logarithmic equation and express irrational solutions in lowest radical form. $$ \log x^{2}=(\log x)^{2} $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=0.9413 $$

Evaluate each logarithmic expression. \(\log _{10} 0.0001\)

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=\frac{3}{5} x+\frac{1}{3} \text { and } g(x)=\frac{5}{3} x-3 $$

The half-life of radium is approximately 1600 years. If the present amount of radium in a certain location is 500 grams, how much will remain after 800 years? Express your answer to the nearest gram.

Graph each of the exponential functions. $$ f(x)=\left(\frac{2}{3}\right)^{x} $$

Approximate each logarithm to three decimal places. $$ \log _{2} 40 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=1.1425 $$

Evaluate each logarithmic expression. \(10^{\log _{10} 5}\)

Determine whether \(f\) and \(g\) are inverse functions. $$ \begin{aligned} &f(x)=x^{2}-3 \text { for } x \geq 0 \text { and } \\ &g(x)=\sqrt{x+3} \text { for } x \geq-3 \end{aligned} $$

Suppose that in a certain culture, the equation \(Q(t)=\) \(1000 e^{0.4 t}\) expresses the number of bacteria present as a function of the time \(t\), where \(t\) is expressed in hours. How many bacteria are present at the end of 2 hours? 3 hours? 5 hours? 2226; 3320; 7389

Graph each of the exponential functions. $$ f(x)=2^{x}-3 $$

Approximate each logarithm to three decimal places. $$ \log _{2} 93 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=2.7619 $$

Evaluate each logarithmic expression. \(10^{\log _{10} 14}\)

Determine whether \(f\) and \(g\) are inverse functions. $$ \begin{array}{lll} f(x)=|x-1| & \text { for } x \geq 1 & \text { and } \\ g(x)=|x+1| & \text { for } x \geq 0 & \underline{\phantom{xxx}} \end{array} $$

The number of bacteria present at a given time under certain conditions is given by the equation \(Q=\) \(5000 e^{0.05 t}\), where \(t\) is expressed in minutes. How many bacteria are present at the end of 10 minutes? \(30 \mathrm{~min}-\) utes? 1 hour? 8244; 22,408; 100,428

Graph each of the exponential functions. $$ f(x)=2^{x}+1 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=4.6873 $$

Evaluate each logarithmic expression. $$ \log _{2}\left(\frac{1}{32}\right) $$

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=\sqrt{x+1} \text { and } g(x)=x^{2}-1 \quad \text { for } x \geq 0 $$

The number of bacteria present in a certain culture after \(t\) hours is given by the equation \(Q=Q_{0} e^{0.3 t}\), where \(Q_{0}\) represents the initial number of bacteria. If 6640 bacteria are present after 4 hours, how many bacteria were present initially? 2000

Graph each of the exponential functions. $$ f(x)=2^{x+2} $$

Approximate each logarithm to three decimal places. $$ \log _{3} 37 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=3.0259 $$

Evaluate each logarithmic expression. $$ \log _{5}\left(\frac{1}{25}\right) $$

Determine whether \(f\) and \(g\) are inverse functions. $$ f(x)=\sqrt{2 x-2} \text { and } g(x)=\frac{1}{2} x^{2}+1 $$

The number of grams \(Q\) of a certain radioactive substance present after \(t\) seconds is given by the equation \(Q=1500 e^{-0.4 t}\). How many grams remain after \(5 \mathrm{sec}-\) onds? 10 seconds? 20 seconds? \(203 ; 27 ; 1\)

Graph each of the exponential functions. $$ f(x)=2^{x-1} $$

Approximate each logarithm to three decimal places. $$ \log _{4} 1.6 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-0.7284 $$

Evaluate each logarithmic expression. $$ \log _{5}\left(\log _{2} 32\right) $$

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=x-4 $$

Graph each of the exponential functions. $$ f(x)=-2^{x} $$

Approximate each logarithm to three decimal places. $$ \log _{4} 3.2 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-1.6246 $$

Evaluate each logarithmic expression. $$ \log _{2}\left(\log _{4} 16\right) $$

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=2 x-1 $$

Suppose that the present population of a city is 75,000 . Using the equation \(P(t)=75,000 e^{0.01 t}\) to estimate future growth, estimate the population (a) 10 years from now, (b) 15 years from now, and (c) 25 years from now. (a) 82,888 (b) 87,138 (c) 96,302