Chapter 5

The number of bacteria in a culture is modeled by the function $$n(t)=500 e^{0.45 t}$$ where \(t\) is measured in hours. (a) What is the initial number of bacteria? (b) What is the relative rate of growth of this bacterium population? Express your answer as a percentage. (c) How many bacteria are in the culture after 3 hours? (d) After how many hours will the number of bacteria reach \(10,000 ?\)

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

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

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

The number of a certain species of fish is modeled by the function $$n(t)=12 e^{0.012 t}$$ where \(t\) is measured in years and \(n(t)\) is measured in millions. (a) What is the relative rate of growth of the fish population? Express your answer as a percentage. (b) What will the fish population be after 5 years? (c) After how many years will the number of fish reach 30 million? (d) Sketch a graph of the fish population function \(n(t) .\)

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

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

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

The fox population in a certain region has a relative growth rate of 8% per year. It is estimated that the population in 2000 was 18,000. (a) Find a function that models the population \(t\) years after 2000. (b) Use the function from part (a) to estimate the fox population in the year 2008. (c) Sketch a graph of the fox population function for the years 2000–2008.

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

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

\(3-8\) Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \log _{5} 25=2} & {\text { (b) } \log _{5} 1=0}\end{array} $$

1–4 ? Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ g(x)=\left(\frac{2}{3}\right)^{x-1} ; \quad g(1.3), g(\sqrt{5}), g(2 \pi), g\left(-\frac{1}{2}\right) $$

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, correct to four decimal places. $$ e^{3 x}=12 $$

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

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

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

The population of a certain city was 112,000 in 1998, and the observed relative growth rate is 4% per year. (a) Find a function that models the population after \(t\) years. (b) Find the projected population in the year 2004. (c) In what year will the population reach 200,000?

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

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

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

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ f(x)=2^{x} $$

The frog population in a small pond grows exponentially. The current population is 85 frogs, and the relative growth rate is 18% per year. (a) Find a function that models the population after \(t\) years. (b) Find the projected population after 3 years. (c) Find the number of years required for the frog population to reach 600.

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

Evaluate the expression. $$ \log _{12} 9+\log _{12} 16 $$

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

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ g(x)=8^{x} $$

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

Evaluate the expression. $$ \log _{2} 6-\log _{2} 15+\log _{2} 20 $$

\(3-8\) Express the equation in exponential form. $$ \begin{array}{ll}{\text { (a) } \ln 5=x} & {\text { (b) } \ln y=5}\end{array} $$

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ f(x)=\left(\frac{1}{3}\right)^{x} $$

A culture contains 1500 bacteria initially and doubles every 30 min. (a) Find a function that models the number of bacteria \(n(t)\) after \(t\) minutes. (b) Find the number of bacteria after 2 hours. (c) After how many minutes will the culture contain 4000 bacteria?

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

Evaluate the expression. $$ \log _{3} 100-\log _{3} 18-\log _{3} 50 $$

\(3-8\) Express the equation in exponential form. $$ (\mathbf{a}) \ln (x+1)=2 \quad \text { (b) } \ln (x-1)=4 $$

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ h(x)=(1.1)^{x} $$

A culture starts with 8600 bacteria. After one hour the count is 10,000. (a) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (b) Find the number of bacteria after 2 hours. (c) After how many hours will the number of bacteria double?

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

Evaluate the expression. $$ \log _{4} 16^{100} $$

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 5^{3}=125} & {\text { (b) } 10^{-4}=0.0001}\end{array} $$

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ g(x)=3 e^{x} $$

The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (d) Find the number of bacteria after 4.5 hours. (e) When will the number of bacteria be 50,000?

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

Evaluate the expression. $$ \log _{2} 8^{33} $$

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 10^{3}=1000} & {\text { (b) } 81^{1 / 2}=9}\end{array} $$

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ h(x)=2 e^{-0.5 x} $$

The population of the world was 5.7 billion in 1995 and the observed relative growth rate was 2% per year. (a) By what year will the population have doubled? (b) By what year will the population have tripled?

Find the solution of the exponential equation, correct to four decimal places. $$ 4+3^{5 x}=8 $$

Evaluate the expression. $$ \log \left(\log 10^{10,000}\right) $$