Chapter 5

Bacteria Culture A certain culture of the bacterium Streptococcus \(A\) initially has 10 bacteria and is observed to double every 1.5 hours. (a) Find an exponential model \(n(t)=n_{0} 2^{t / 2}\) for the number of bacteria in the culture after \(t\) hours. (b) Estimate the number of bacteria after 35 hours. (c) When will the bacteria count reach \(10,000 ?\)

\(\log x\) is the exponent to which the base 10 must be raised to get ________.So we can complete the following table for log \(x\) $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {10^{3}} & {10^{2}} & {10^{1}} & {10^{6}} & {10^{-1}} & {10^{-2}} & {10^{-3}} & {10^{1 / 2}} \\ \hline \log x & {} & {} & {} & {} & {} \\ \hline\end{array} $$

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 ln of each side to get the equivalent equation ____________. (c) Now we use a calculator to find \(x=\) ____________.

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

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

The function \(f(x)=\log _{8} x\) is the logarithm function with base ________, So \(f(9)=\) _________ ,\(f(1)=\) _______,\(f(t)=\) _________, \(f(81)=\) __________, and \(f(3)=\) _________

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 \(x=\) __________.

Squirrel Population A grey squirrel population was in- troduced 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) \(5^{3}=125,\) so log ___ = ___ (b) \(\log _{5} 25=2,\) so__ = ___

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

The logarithm of a number raised to a power is the same as the power _____ the logarithm of the number. So \(\log _{5}\left(25^{10}\right)=\) _____ . _____.

\(3-4\) . Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ h(x)=e^{x} ; \quad h(3), h(0.23), h(1), h(-2) $$

(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.

Bird Population 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.

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

(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 _____.

In the formula \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) for compound interest the letters \(P, r, n,\) and \(t\) stand for _____, _____, _____, and _____, respectively, and \(A(t)\) stands for _____. So if \(\$ 100\) is invested at an interest rate of 6\(\%\) compounded quarterly, then the amount after 2 years is _____.

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. $$ f(x)=4^{x} ; \quad f(0.5), f(\sqrt{2}), f(-\pi), f\left(\frac{1}{3}\right) $$

\(5-6\) mplete the table of values, rounded to two decimal places, and sketch a graph of the function. $$ \begin{array}{|c|c|}\hline x & {f(x)=3 e^{x}} \\ \hline-2 & {} \\ {-1} & {} \\\ {-0.5} \\ {0} \\ {0} \\ {0.5} \\ {1} \\ {2} \\ \hline\end{array} $$

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. $$ f(x)=3^{x+1} ; \quad f(-1.5), f(\sqrt{3}), f(e), f\left(-\frac{5}{4}\right) $$

\(5-6\) mplete the table of values, rounded to two decimal places, and sketch a graph of the function. $$ \begin{array}{|c|c|}\hline x & {f(x)=2 e^{-0.5 x}} \\ \hline-3 & {} \\ {-2} & {} \\ {0} \\ {0} \\ {1} \\ {2} \\ {3} \\ \hline\end{array} $$

Population of a Country 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.

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{s} 25=2} & {\text { (b) } \log _{s} 1=0}\end{array} $$

\(7-18\) Evaluate the expression. $$ \log _{3} \sqrt{27} $$

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

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

Bacteria Culture It is observed that a certain bacteria culture has a relative growth rate of 12\(\%\) per hour, but in the 0presence 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 \(5 \% .\)

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{10} 0.1=-1} & {\text { (b) } \log _{8} 512=3}\end{array} $$

\(7-18\) Evaluate the expression. $$ \log _{2} 160-\log _{2} 5 $$

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

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

Population of a City The population of a certain city was \(112,000\) in 2006 , and the observed doubling time for the population is 18 years. (a) Find an exponential model \(n(t)=n_{0} 2^{l a}\) for the population \(t\) years after 2006 . (b) Find an exponential model \(n(t)=n_{0} e^{r t}\) for the population \(t\) years after 2006 . (c) Sketch a graph of the population at time \(t\) (d) Estimate when the population will reach \(500,000\) .

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{8} 2=\frac{1}{3}} & {\text { (b) } \log _{2}\left(\frac{1}{4}\right)=-3}\end{array} $$

\(7-18\) Evaluate the expression. $$ \log 4+\log 25 $$

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

\(7-14\) . 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. $$ y=e^{-x}-1 $$

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

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{3} 81=4} & {\text { (b) } \log _{8} 4=\frac{2}{3}}\end{array} $$

\(7-18\) Evaluate the expression. $$ \log \frac{1}{\sqrt{1000}} $$

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. $$ g(x)=8^{x} $$

Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \ln 5=x} & {\text { (b) } \ln y=5}\end{array} $$

\(7-18\) Evaluate the expression. $$ \log _{4} 192-\log _{4} 3 $$

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

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

Express the equation in exponential form. $$ \ln (x+1)=2 \quad \text { (b) } \ln (x-1)=4 $$

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

\(7-18\) Evaluate the expression. $$ \log _{12} 9+\log _{12} 16 $$