Chapter 5

Solve the equation. $$ \log x-\log (x+1)=3 \log 4 $$

U.S. population growth The 1980 population of the United States was approximately 231 million, and the population has been growing continuously at a rate of \(1.03 \%\) per year. Predict the population \(N(t)\) in the year 2020 if this growth trend continues.

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 5^{x}+125\left(5^{-x}\right)=30 $$

Exer. 19-34: Solve the equation. $$ \log _{5}(x-2)=\log _{5}(3 x+7) $$

Exer. 21-24: Determine the domain and range of \(f^{-1}\) for the given function without actually finding \(f^{-1}\). Hint: First find the domain and range of \(f\). $$ f(x)=-\frac{2}{x-1} $$

Sketch the graph of \(f\). $$f(x)=3^{1-x^{2}}$$

Solve the equation. $$ \log (x+2)-\log x=2 \log 4 $$

Population growth in India The 1985 population estimate for India was 766 million, and the population has been growing continuously at a rate of about \(1.82 \%\) per year. Assuming that this rapid growth rate continues, estimate the population \(N(t)\) of India in the year \(2015 .\)

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 3\left(3^{x}\right)+9\left(3^{-x}\right)=28 $$

Exer. 19-34: Solve the equation. $$ \log _{7}(x-5)=\log _{7}(6 x) $$

Exer. 21-24: Determine the domain and range of \(f^{-1}\) for the given function without actually finding \(f^{-1}\). Hint: First find the domain and range of \(f\). $$ f(x)=\frac{5}{x+3} $$

Sketch the graph of \(f\). $$f(x)=2^{-(x+1)^{2}}$$

Solve the equation. $$ \ln (-4-x)+\ln 3=\ln (2-x) $$

Longevity of halibut In fishery science, a cohort is the collection of fish that results from one annual reproduction. It is usually assumed that the number of fish \(N(t)\) still alive after \(t\) years is given by an exponential function. For Pacific halibut, \(N(t)=N_{0} e^{-0.2 t}\), where \(N_{0}\) is the initial size of the cohort. Approximate the percentage of the original number still alive after 10 years.

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 4^{x}-3\left(4^{-x}\right)=8 $$

Exer. 21-24: Determine the domain and range of \(f^{-1}\) for the given function without actually finding \(f^{-1}\). Hint: First find the domain and range of \(f\). $$ f(x)=\frac{4 x+5}{3 x-8} $$

Sketch the graph of \(f\). $$f(x)=3^{x}+3^{-x}$$

Solve the equation. $$ \ln x+\ln (x+6)=\frac{1}{2} \ln 9 $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 2^{x}-6\left(2^{-x}\right)=6 $$

Exer. 19-34: Solve the equation. $$ \ln x^{2}=\ln (12-x) $$

Exer. 21-24: Determine the domain and range of \(f^{-1}\) for the given function without actually finding \(f^{-1}\). Hint: First find the domain and range of \(f\). $$ f(x)=\frac{2 x-7}{9 x+1} $$

Sketch the graph of \(f\). $$f(x)=3^{x}-3^{-x}$$

Solve the equation. $$ \log _{2}(x+7)+\log _{2} x=3 $$

Exer. 25-32: Solve the equation without using a calculator. $$ \log \left(x^{2}\right)=(\log x)^{2} $$

Exer. 19-34: Solve the equation. $$ \log _{3}(x-4)=2 $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=3 x+5 $$

Solve the equation. $$ \log _{6}(x+5)+\log _{6} x=2 $$

Exer. 25-32: Solve the equation without using a calculator. $$ \log \sqrt{x}=\sqrt{\log x} $$

Exer. 19-34: Solve the equation. $$ \log _{2}(x=5)=4 $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=7-2 x $$

Solve the equation. $$ \log _{3}(x+3)+\log _{3}(x+5)=1 $$

Exer. 25-32: Solve the equation without using a calculator. $$ \log (\log x)=2 $$

Exer. 19-34: Solve the equation. $$ \log _{9} x=\frac{3}{2} $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\frac{1}{3 x-2} $$

Solve the equation. $$ \log _{3}(x-2)+\log _{3}(x-4)=2 $$

Exer. 25-32: Solve the equation without using a calculator. $$ \log \sqrt{x^{3}-9}=2 $$

Exer. 19-34: Solve the equation. $$ \log _{4} x=-\frac{3}{2} $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\frac{1}{x+3} $$

Solve the equation. $$ \log (x+3)=1-\log (x-2) $$

Growth of children The Jenss model is generally regarded as the most accurate formula for predicting the height of preschool children. If \(y\) is height (in centimeters) and \(x\) is age (in years), then $$ y=79.041+6.39 x-e^{3.261-0.993 x} $$ for \(\frac{1}{4} \leq x \leq 6\). From calculus, the rate of growth \(R\) (in \(\mathrm{cm} /\) year) is given by \(R=6.39+0.993 e^{3261-0.993 x}\). Find the height and rate of growth of a typical 1-year-old child.

Exer. 25-32: Solve the equation without using a calculator. $$ x^{\sqrt{\log x}}=10^{8} $$

Exer. 19-34: Solve the equation. $$ \ln x^{2}=-2 $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\frac{3 x+2}{2 x-5} $$

Find an exponential function of the form \(f(x)=b a^{x}\) that has the given \(y\)-intercept and passes through the point \(P\). \(y\)-intercept \(8 ; \quad P(3,1)\)

Solve the equation. $$ \log (57 x)=2+\log (x-2) $$

Particle velocity A very small spherical particle (on the order of 5 microns in diameter) is projected into still air with an initial velocity of \(v_{0} \mathrm{~m} / \mathrm{sec}\), but its velocity decreases because of drag forces. Its velocity \(t\) seconds later is given by \(v(t)=v_{0} e^{-a t}\) for some \(a>0\), and the distance \(s(t)\) the particle travels is given by $$ s(t)=\frac{v_{0}}{a}\left(1-e^{-a t}\right) . $$ The stopping distance is the total distance traveled by the particle. (a) Find a formula that approximates the stopping distance in terms of \(v_{0}\) and \(a\). (b) Use the formula in part (a) to estimate the stopping distance if \(v_{0}=10 \mathrm{~m} / \mathrm{sec}\) and \(a=8 \times 10^{5}\).

Exer. 25-32: Solve the equation without using a calculator. $$ \log \left(x^{3}\right)=(\log x)^{3} $$

Exer. 19-34: Solve the equation. $$ \log x^{2}=-4 $$

Exer. 25-42: Find the inverse function of \(f\). $$ f(x)=\frac{4 x}{x-2} $$

Find an exponential function of the form \(f(x)=b a^{x}\) that has the given \(y\)-intercept and passes through the point \(P\). y-intercept 6; \(\quad P\left(2, \frac{3}{32}\right)\)