Chapter 5

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

Decide which of the two plans will provide a better yield. (Interest rates stated are annual rates.) Plan A: \(\$ 40,000\) invested for 3 years at \(2.5 \%,\) compounded quarterly Plan B: \(\$ 40,000\) invested for 3 years at \(2.4 \%,\) compounded continuously

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

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=5 x^{3}-7$$

$$\text { Solve each formula for the indicated variable.}$$ $$A=T_{0}+C e^{-k}, \text { for } k$$

Decide which of the two plans will provide a better yield. (Interest rates stated are annual rates.) Plan A: \(\$ 50,000\) invested for 10 years at \(4.75 \%,\) compounded daily \((n=365)\) Plan B: \(\$ 50,000\) invested for 10 years at \(4.7 \%,\) compounded continuously

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{x^{5} y^{3}}{3}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=4-3 x^{3}$$

$$\text { Solve each formula for the indicated variable.}$$ $$y=\frac{K}{1+a e^{-b x}}, \text { for } b$$

Use the table capabilities of your calculator to work Comparison of Two Accounts You have the choice of investing \(\$ 1000\) at an annual rate of \(5 \%,\) compounded either annually or monthly. Let \(Y_{1}\) represent the investment compounded annually, and let \(Y_{2}\) represent the investment compounded monthly. Graph both \(Y_{1}\) and \(Y_{2}\), and observe the slight differences in the curves. Then use a table to compare the graphs numerically. What is the difference between the returns for the investments after 1 year, 2 years, 5 years, 10 years, 20 years, 30 years, and 40 years?

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

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{x}{4+3 x}$$

$$\text { Solve each formula for the indicated variable.}$$ $$y=A+B\left(1-e^{-c x}\right), \text { for } x$$

Use the table capabilities of your calculator to work Comparison of Two Accounts You have the choice of investing \(\$ 1000\) at an annual rate of \(7.5 \%\) compounded daily or \(7.75 \%\) compounded annually. Let \(Y_{1}\) represent the investment at \(7.5 \%\) compounded daily, and let \(Y_{2}\) represent the investment at \(7.75 \%\) compounded annually. Graph both \(Y_{1}\) and \(Y_{2}\), and observe the slight differences in the curves. Then use a table with \(Y_{3}=Y_{1}-Y_{2}\) to compare the graphs numerically. What is the difference between the returns for the investments after 1 year, 2 years, 5 years, 10 years, 20 years, 30 years, and 40 years? Why does the lower interest rate yield the greater return over time?

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

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

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

Solve each problem. Atmospheric Pressure The atmospheric pressure (in millibars) at a given altitude (in meters) is shown in the table. $$\begin{array}{c|c|c|c} \text { Altitude } & \text { Pressure } & \text { Altitude } & \text { Pressure } \\ \hline 0 & 1013 & 6000 & 472 \\ 1000 & 899 & 7000 & 411 \\ 2000 & 795 & 8000 & 357 \\ 3000 & 701 & 9000 & 308 \\ 4000 & 617 & 10,000 & 265 \\ 5000 & 541 & & \end{array}$$ (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 $$ P(x)=1013 e^{-0.0001341 x} $$ approximates the data. Use a graphing calculator to graph \(P\) and the data on the same coordinate axes. (d) Use \(P\) 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.

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

$$\text { Solve each formula for the indicated variable.}$$ $$\log A=\log B-C \log x, \text { for } A$$

Solve each problem. World Population Growth since 2000 , world population in millions closely fits the exponential function $$ y=6079 e^{0.0126 x} $$ where \(x\) is the number of years since 2000 . (Image can't copy) (a) The world population was about 6555 million in 2006 . How closely does the function approximate this value? (b) Use this model to estimate the population in 2010 . (c) Use this model to predict the population in 2025 . (d) Explain why this model may not be accurate for 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. $$\left(\log _{b} k-\log _{b} m\right)-\log _{b} a$$

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

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

Traffic Flow \(\quad\) 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 within a 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. $$2 \log _{m} a-3 \log _{m} b^{2}$$

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f(2)$$

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

Growth of E. coli Bacteria \(\mathrm{A}\) 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. $$\frac{1}{2} \log _{y} p^{3} q^{4}-\frac{2}{3} \log _{y} p^{4} q^{3}$$

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f(0)$$

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

Assume that \(f(x)=a^{x},\) where \(a>1\) Is \(f\) a one-to-one function? If so, based on Section 5.1 what kind of related function exists for \(f ?\)

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

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f(-2)$$

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

Assume that \(f(x)=a^{x},\) where \(a>1\) 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. $$\log _{b}(2 y+5)-\frac{1}{2} \log _{b}(y+3)$$

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f^{-1}(8)$$

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

Assume that \(f(x)=a^{x},\) where \(a>1\) If \(f^{-1}\) exists, find an equation for \(y=f^{-1}(x),\) using the method described in Section \(5.1 .\) (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. $$-\frac{2}{3} \log _{5} 5 m^{2}+\frac{1}{2} \log _{5} 25 m^{2}$$

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

Assume that \(f(x)=a^{x},\) where \(a>1\) 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. $$-\frac{3}{4} \log _{3} 16 p^{4}-\frac{2}{3} \log _{3} 8 p^{3}$$

Let \(f(x)=x^{3} .\) Evaluate each expression. $$f^{-1}(-8)$$

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

Assume that \(f(x)=a^{x},\) where \(a>1\) 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. $$3 \log x-4 \log y$$