Chapter 10

Explain how to determine \(\log _{4} 76\) without using Property 10.9.

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 2}{0.03} $$

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} 80\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=x^{2}+1 $$

Graph \(f(x)=e^{x}\). Where should the graphs of \(f(x)=\) \(e^{x-4}, f(x)=e^{x-6}\), and \(f(x)=e^{x+5}\) be located? Graph all three functions on the same set of axes with \(f(x)=e^{x}\).

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\log 2}{5 \log 1.02} $$

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} 55\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=x^{3} $$

Graph \(f(x)=e^{x}\). Now predict the graphs for \(f(x)=\) \(-e^{x}, f(x)=e^{-x}\), and \(f(x)=-e^{-x}\). Graph all three functions on the same set of axes with \(f(x)=e^{x}\).

Explain how you would solve the equation \(2^{x}=64\) and also how you would solve the equation \(2^{x}=53\).

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\log 5}{3 \log 1.07} $$

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}\left(\frac{5}{11}\right)\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=-3 x+1 $$

How do you think the graphs of \(f(x)=e^{x}, f(x)=e^{2 x}\), and \(f(x)=2 e^{x}\) will compare? Graph them on the same set of axes to see if you were correct.

How do logarithms with a base of 9 compare to logarithms with a base of 3 ? Explain how you reached this conclusion.

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} 25\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=(x-3)^{2}+1 $$

Find an approximate solution, to the nearest hundredth, for each of the following equations by graphing the appropriate function and finding the \(x\) intercept. (a) \(e^{x}=7\) (b) \(e^{x}=21\) (c) \(e^{x}=53\) (d) \(2 e^{x}=60\) (e) \(e^{x+1}=150\) (f) \(e^{x-2}=300\)

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} \sqrt{11}\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=-(x+2)^{2}-1 $$

Graph \(f(x)=x, f(x)=e^{x}\), and \(f(x)=\ln x\) on the same set of axes.

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}(5)^{2 / 3}\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=x^{2}-2 x+6 $$

How long will it take \(\$ 500\) to be worth \(\$ 1500\) if it is invested at \(7.5 \%\) interest compounded semiannually?

Solve the equation \(\frac{5^{x}-5^{-x}}{2}=3\). Express your answer to the nearest hundredth. \(\\{1.13\\}\)

Graph \(f(x)=x, f(x)=10^{x}\), and \(f(x)=\log x\) on the same set of axes.

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} 88\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=-2 x^{2}-16 x-35 $$

How long will it take \(\$ 5000\) to triple if it is invested at \(6.75 \%\) interest compounded quarterly?

$$ \text { Solve the equation } y=\frac{10^{x}+10^{-x}}{2} \text { for } x \text { in terms of } y \text {. } $$

Graph \(f(x)=\ln x\). How should the graphs of \(f(x)=2\) \(\ln x, f(x)=4 \ln x\), and \(f(x)=6 \ln x\) compare to the graph of \(f(x)=\ln x\) ? Graph the three functions on the same set of axes with \(f(x)=\ln x\).

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8} 320\)

Find the intervals on which the given function is increasing and the intervals on which it is decreasing. $$ f(x)=x^{2}+3 x-1 $$

$$ \text { Solve the equation } y=\frac{e^{x}-e^{-x}}{2} \text { for } x \text { in terms of } y \text {. } $$

Graph \(f(x)=\log x\). Now predict the graphs for \(f(x)=\) \(2+\log x, f(x)=-2+\log x\), and \(f(x)=-6+\log x\). Graph the three functions on the same set of axes with \(f(x)=\log x\).

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}\left(\frac{25}{11}\right)\)

Does the function \(f(x)=4\) have an inverse? Explain your answer.

Graph \(\ln x\). Now predict the graphs for \(f(x)=\ln (x-2)\), \(f(x)=\ln (x-6)\), and \(f(x)=\ln (x+4)\). Graph the three functions on the same set of axes with \(f(x)=\ln x\).

Given that \(\log _{8} 5=0.7740\) and \(\log _{8} 11=1.1531\), evaluate each expression using Properties \(10.5-10.7\) \(\log _{8}\left(\frac{121}{25}\right)\)

Graph \(f(x)=x, f(x)=2^{x}\), and \(f(x)=\log _{2} x\) on the same set of axes.

For each of the following, (a) predict the general shape and location of the graph, and (b) use your graphing calculator to graph the function to check your prediction. (a) \(f(x)=\log x+\ln x\) (b) \(f(x)=\log x-\ln x\) (c) \(f(x)=\ln x-\log x\) (d) \(f(x)=\ln x^{2}\)

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x y z $$

Are the functions \(f(x)=x^{4}\) and \(g(x)=\sqrt[4]{x}\) inverses of each other? Explain your answer.

Graph \(f(x)=x, f(x)=(0.5)^{x}\), and \(f(x)=\log _{0.5} x\) on the same set of axes.

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{h} 5 x $$

Graph \(f(x)=\log _{2} x\). Now predict the graphs for \(f(x)=\) \(\log _{3} x, f(x)=\log _{4} x\), and \(f(x)=\log _{8} x\). Graph these three functions on the same set of axes with $$ f(x)=\log _{2} x \text {. } $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{y}{z}\right) $$

The function notation and the operation of composition can be used to find inverses as follows: To find the inverse of \(f(x)=5 x+3\), we know that \(f\left(f^{-1}(x)\right)\) must produce \(x\). Therefore $$ \begin{aligned} f\left(f^{-1}(x)\right)=5\left[f^{-1}(x)\right]+3 &=x \\ 5\left[f^{-1}(x)\right] &=x-3 \\ f^{-1}(x) &=\frac{x-3}{5} \end{aligned} $$

Graph \(f(x)=\log _{5} x\). Now predict the graphs for \(f(x)=\) \(2 \log _{5} x, f(x)=-4 \log _{5} x\), and \(f(x)=\log _{5}(x+4)\). Graph these three functions on the same set of axes with \(f(x)=\log _{5} x\).

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(\frac{x^{2}}{y}\right) $$