Chapter 4

Let's solve the exponential equation \(2 e^{x}=50\). (a) First, we isolate \(e^{x}\) to get the equivalent equation _____. (b) Next, we take In of each side to get the equivalent equation _____.

The function \(f(x)=5^{x}\) is an exponential function with base \(_____\) \(=; f(-2)=\) \(______\) \(f(0)=\) \(_____\) \(f(2)=\)and \(f(6)=\) \(_____\)

The function \(f(x)=e^{x}\) is called the ________ function. The number \(e\) is approximately equal to ________.

\(\log x\) is the exponent to which the base 10 must be raised to get _______. So we can complete the following table for \(\log x\). $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & 10^{3} & 10^{2} & 10^{1} & 10^{0} & 10^{-1} & 10^{-2} & 10^{-3} & 10^{1 / 2} \\ \hline \log x & & & & & & & & \\ \hline \end{array}$$

The logarithm of a product of two numbers is the same as the ______ of the logarithms of these numbers. So $$\log _{5}(25 \cdot 125)=$$ _____ + ______ .

Let's solve the logarithmic equation \(\log 3+\log (x-2)=\log x\). (a) First, we combine the logarithms to get the equivalent equation _____ (b) Next, we write each side in exponential form to get the equivalent equation _____. (c) Now we find \(x=\) _____.

These exercises use the population growth model. A certain culture of the bacterium Rhodobacter sphaeroides initially has 25 bacteria and is observed to double every 5 hours. (a) Find an exponential model \(n(t)=n_{0} 2^{\prime / a}\) for the number of bacteria in the culture after \(t\) hours. (b) Estimate the number of bacteria after 18 hours. (c) After how many hours will the bacteria count reach 1 million?

The function \(f(x)=\log _{9} x\) is the logarithm function with base ______. So \(f(9)=\) ______, \(f(1)=\) ______, \(f\left(\frac{1}{9}\right)=\) ______, \(f(81)=\) ______, and \(f(3)=\) ______.

The logarithm of a quotient of two numbers is the same as the _______ of the logarithms of these numbers. So \(\log _{5}\left(\frac{25}{125}\right)=\) ________ - ________.

Find the solution of the exponential equation, rounded to four decimal places. $$10^{x}=25$$

These exercises use the population growth model. A grey squirrel population was introduced in a certain county of Great Britain 30 years ago. Biologists observe that the population doubles every 6 years, and now the population is \(100,000\). (a) What was the initial size of the squirrel population? (b) Estimate the squirrel population 10 years from now. (c) Sketch a graph of the squirrel population.

(a) To obtain the graph of \(g(x)=2^{x}-1,\) we start with the graph of \(f(x)=2^{x}\) and shift it \(______\) (upward/downward) 1 unit. (b) To obtain the graph of \(h(x)=2^{x-1}\), we start with the graph of \(f(x)=2^{x}\) and shift it to the \(_____\) (left/right) 1 unit.

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$h(x)=e^{x} ; \quad h(3), h(0.23), h(1), h(-2)$$

Find the solution of the exponential equation, rounded to four decimal places. $$10^{-x}=4$$

These exercises use the population growth model. A certain species of bird was introduced in a certain county 25 years ago. Biologists observe that the population doubles every 10 years, and now the population is \(13,000\). (a) What was the initial size of the bird population? (b) Estimate the bird population 5 years from now. (c) Sketch a graph of the bird population.

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$h(x)=e^{-2 x} ; \quad h(1), h(\sqrt{2}), h(-3), h\left(\frac{1}{2}\right)$$

(a) We can expand \(\log \left(\frac{x^{2} y}{z}\right)\) to get ______________. (b) We can combine \(2 \log x+\log y-\log z\) to get _______________.

Find the solution of the exponential equation, rounded to four decimal places. $$e^{-2 x}=7$$

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$f(x)=4^{x} ; \quad f(0.5), f(\sqrt{2}), f(-\pi), f\left(\frac{1}{3}\right)$$

Complete the table of values, rounded to two decimal places, and sketch a graph of the function. $$\begin{array}{|c|c|} \hline x & f(x)=3 e^{x} \\ \hline-2 & \\ -1 & \\ -0.5 & \\ 0 & \\ 0.5 & \\ 1 & \\ 2 & \\ \hline \end{array}$$

Find the solution of the exponential equation, rounded to four decimal places. $$e^{3 x}=12$$

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$f(x)=3^{x+1} ; \quad f(-1.5), f(\sqrt{3}), f(e), f\left(-\frac{5}{4}\right)$$

Complete the table of values, rounded to two decimal places, and sketch a graph of the function. $$\begin{array}{|c|c|} \hline x & f(x)=2 e^{-25 x} \\ \hline-3 & \\ -2 & \\ -1 & \\ 0 & \\ 1 & \\ 2 & \\ 3 & \\ \hline \end{array}$$

Express the equation in exponential form. (a) \(\log _{5} 25=2\) (b) \(\log _{5} 1=0\)

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$g(x)=\left(\frac{2}{3}\right)^{x-1} ; \quad g(1.3), g(\sqrt{5}), g(2 \pi), g\left(-\frac{1}{2}\right)$$

These exercises use the population growth model. The population of a country has a relative growth rate of \(3 \%\) per year. The government is trying to reduce the growth rate to \(2 \% .\) The population in 1995 was approximately 110 million. Find the projected population for the year 2020 for the following conditions. (a) The relative growth rate remains at \(3 \%\) per year. (b) The relative growth rate is reduced to \(2 \%\) per year.

Find the solution of the exponential equation, rounded to four decimal places. $$2^{1-x}=3$$

Evaluate the expression. $$\log _{3} \sqrt{27}$$

Graph the function, not by plotting points, but by starting from the graph of \(y=e^{x}\) in Figure \(1 .\) State the domain, range, and asymptote. $$f(x)=-e^{x}$$

Express the equation in exponential form. (a) \(\log _{10} 0.1=-1\) (b) \(\log _{8} 512=3\)

Find the solution of the exponential equation, rounded to four decimal places. $$3^{2 x-1}=5$$

These exercises use the population growth model. It is observed that a certain bacteria culture has a relative growth rate of \(12 \%\) per hour, but in the presence of an antibiotic the relative growth rate is reduced to \(5 \%\) per hour. The initial number of bacteria in the culture is 22. Find the projected population after 24 hours for the following conditions. (a) No antibiotic is present, so the relative growth rate is \(12 \%\). (b) An antibiotic is present in the culture, so the relative growth rate is reduced to \(5 \%\).

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$g(x)=\left(\frac{3}{4}\right)^{2 x} ; \quad g(0.7), g(\sqrt{7} / 2), g(1 / \pi), g\left(\frac{2}{3}\right)$$

Evaluate the expression. $$\log _{2} 160-\log _{2} 5$$

Graph the function, not by plotting points, but by starting from the graph of \(y=e^{x}\) in Figure \(1 .\) State the domain, range, and asymptote. $$y=1-e^{x}$$

Express the equation in exponential form. (a) \(\log _{8} 2=\frac{1}{3}\) (b) \(\log _{2}\left(\frac{1}{8}\right)=-3\)

Find the solution of the exponential equation, rounded to four decimal places. $$3 e^{x}=10$$

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$f(x)=2^{x}$$

Evaluate the expression. $$\log 4+\log 25$$

Graph the function, not by plotting points, but by starting from the graph of \(y=e^{x}\) in Figure \(1 .\) State the domain, range, and asymptote. $$y=e^{-x}-1$$

Express the equation in exponential form. (a) \(\log _{3} 81=4\) (b) \(\log _{8} 4=\frac{2}{3}\)

Find the solution of the exponential equation, rounded to four decimal places. $$2 e^{12 x}=17$$

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$g(x)=8^{x}$$

Evaluate the expression. $$\log \frac{1}{\sqrt{1000}}$$

Find the solution of the exponential equation, rounded to four decimal places. $$e^{1-4 x}=2$$

Express the equation in exponential form. (a) \(\ln 5=x\) (b) \(\ln y=5\)

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$f(x)=\left(\frac{1}{3}\right)^{x}$$

Evaluate the expression. $$\log _{4} 192-\log _{4} 3$$

Find the solution of the exponential equation, rounded to four decimal places. $$4\left(1+10^{5 x}\right)=9$$

Express the equation in exponential form. (a) \(\ln (x+1)=2\) (b) \(\ln (x-1)=4\)