Chapter 6

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{x}{1-3 x}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{p} \sqrt[3]{\frac{m^{5}}{k t^{2}}}$$

Solve each formula for the indicated variable. $$m=6-2.5 \log \left(\frac{M}{M_{0}}\right), \text { for } M$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{3-x}{2 x+1}$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\log _{a} x+\log _{a} y-\log _{a} m$$

The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{|c|c|c|c|} \hline \text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\ 1000 & 899 & 7000 & 411 \\ 2000 & 795 & 8000 & 357 \\ 3000 & 701 & 9000 & 308 \\ 4000 & 617 & 10,000 & 265 \\ 5000 & 541 & & \\ \hline \end{array}$$ (a) Use a graphing calculator to make a scatter diagram of the data for atmospheric pressure \(P\) at altitude \(x\). (b) Use the concept of average rate of change to determine whether a linear or exponential function would fit the data better. (c) The function $$P(x)=1013 e^{-00001341 x}$$ approximates the data. Use a graphing calculator to graph \(P\) and the data on the same coordinate axes. (d) Use function \(P\) to predict the pressures at \(1500 \mathrm{m}\) and \(11,000 \mathrm{m},\) and compare them with the actual values of 846 millibars and 227 millibars, respectively.

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{2 x+1}{x-1}$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\left(\log _{b} k-\log _{b} m\right)-\log _{b} a$$

Solve each formula for the indicated variable. $$d=10 \log \left(\frac{I}{I_{0}}\right), \text { for } I$$

Between 2000 and 2017 , world population in millions is modeled by the exponential function $$f(x)=6136 e^{001206 x}$$ where \(x\) is the number of years since \(2000 .\) (Source: www. worldometers.info) (a) The world population was about 6763 million in 2008 . How closely does the function approximate this value? (b) Use this model to estimate the population in 2018 . (c) Use this model to predict the population in 2025 .

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$2 \log _{m} a-3 \log _{m} b^{2}$$

Solve each formula for the indicated variable. $$A=P\left(1+\frac{r}{n}\right)^{n t}, \text { for } t$$

At an intersection, cars arrive randomly at an average rate of 30 cars per hour. Using the function $$f(x)=1-e^{-0.5 x}$$ highway engineers estimate the likelihood or probability that at least one car will enter the intersection withina period of \(x\) minutes. (Source: Mannering. F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) (a) Evaluate \(f(2)\) and interpret the answer. (b) Graph \(f\) for \(0 \leq x \leq 60\). What happens to the likelihood that at least one car enters the intersection during a 60 -minute period?

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\frac{1}{2} \log _{y} p^{3} q^{4}-\frac{2}{3} \log _{y} p^{4} q^{3}$$

Solve each formula for the indicated variable. $$D=160+10 \log x, \text { for } x$$

Growth of \(\mathbf{E}\) coli Bacteria \(\mathbf{A}\) type of bacteria that inhabits the intestines of animals is named \(E\). coli (Escherichia coli). These bacteria are capable of rapid growth and can be dangerous to humans- especially children. In one study, \(E\) coli bacteria were capable of doubling in number every 49.5 minutes. Their number after \(x\) minutes can be modeled by the function $$ N(x)=N_{0} e^{0.014 x} $$ (Source: Stent, G. S... Molecular Biology of Bacterial Viruses, W. H. Freeman.) Suppose \(N_{0}=500,000\) is the initial number of bacteria per milliliter. (a) Make a conjecture about the number of bacteria per milliliter after 99 minutes. Verify your conjecture. (b) Estimate graphically the time elapsed until there were 25 million bacteria per milliliter.

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$2 \log _{a}(z-1)+\log _{a}(3 z+2), z>1$$

The given equations are quadratic in form. Solve each and give exact solutions. $$e^{2 x}-6 e^{x}+8=0$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\log _{b}(2 y+5)-\frac{1}{2} \log _{b}(y+3)$$

The given equations are quadratic in form. Solve each and give exact solutions. $$e^{2 x}-8 e^{x}+15=0$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(f\) has an inverse function \(f^{-1},\) sketch \(f\) and \(f^{-1}\) on the same axes.

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$-\frac{2}{3} \log _{5} 5 m^{2}+\frac{1}{2} \log _{5} 25 m^{2}$$

The given equations are quadratic in form. Solve each and give exact solutions. $$2 e^{2 x}+e^{x}=6$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(f^{-1}\) exists, find an equation for \(y=f^{-1}(x),\) using the method described earlier in this chapter. (You need not solve for \(y .)\)

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$-\frac{3}{4} \log _{3} 16 p^{4}-\frac{2}{3} \log _{3} 8 p^{3}$$

The given equations are quadratic in form. Solve each and give exact solutions. $$3 e^{2 x}+2 e^{x}=1$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(a=10,\) what is an equation for \(y=f^{-1}(x) ?\) (You need not solve for \(y .\) )

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$3 \log x\(\log \frac{x^{3}}{y^{4}}\)-4 \log y$$

The given equations are quadratic in form. Solve each and give exact solutions. $$\frac{1}{2} e^{2 x}+e^{x}=1$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If \(a=e,\) what is an equation for \(y=f^{-1}(x) ?\) (You need not solve for \(y .\) )

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\frac{1}{2} \log x-\frac{1}{3} \log y-2 \log z$$

The given equations are quadratic in form. Solve each and give exact solutions. $$\frac{1}{4} e^{2 x}+2 e^{x}=3$$

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If the point \((p, q)\) is on the graph of \(f\), then the point _______ is on the graph of \(f^{-1}\).

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\ln (a+b)+\ln a-\frac{1}{2} \ln 4$$

The given equations are quadratic in form. Solve each and give exact solutions. $$3^{2 x}+35=12\left(3^{x}\right)$$

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1 . Assume that all variables represent positive real numbers. $$\frac{4}{3} \ln m-\frac{2}{3} \ln 8 n-\ln m^{3} n^{2}$$

The given equations are quadratic in form. Solve each and give exact solutions. $$5^{2 x}+3\left(5^{x}\right)=28$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5} 10$$

The given equations are quadratic in form. Solve each and give exact solutions. $$\left(\log _{2} x\right)^{2}+\log _{2} x=2$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{9} 12$$

The given equations are quadratic in form. Solve each and give exact solutions. $$(\log x)^{2}-6 \log x=7$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{15} 5$$

The given equations are quadratic in form. Solve each and give exact solutions. $$(\ln x)^{2}+16=10 \ln x$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{1 / 2} 3$$

The given equations are quadratic in form. Solve each and give exact solutions. $$2(\ln x)^{2}+9 \ln x=5$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{100} 83$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=-2 e^{x}+5$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{200} 175$$

Solve the equation \(f(x)=0\) analytically and then use a graph of \(y=f(x)\) to solve the inequalities \(f(x)<0\) and \(f(x) \geq 0\). $$f(x)=-3 e^{x}+7$$

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{29} 7.5$$