Chapter 4

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 5 \ln (2 x)=20 $$

Describe a difference between exponential growth and logistic growth.

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 3 \ln x+5 \ln y-6 \ln z $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 6 \ln (2 x)=30 $$

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 4 \ln x+7 \ln y-3 \ln z $$

The figure shows the graph of \(f(x)=\log x .\) Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=1-\log x $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 6+2 \ln x=5 $$

You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{2}(\log x+\log y) $$

The figure shows the graph of \(f(x)=\log x .\) Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=2-\log x $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7+3 \ln x=6 $$

Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=\ln (x+2) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 6+2 \ln x=5 $$

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

Use a calculator with a \(\left[\mathrm{y}^{x}\right]\) key or \(a[\wedge]\) key to solve Exercises \(65-70\) India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential function \(f(x)-574(1.026)^{x}\) models the population of India, \(f(x),\) in millions, \(x\) years after 1974 . a. Substitute 0 for \(x\) and, without using a calculator, find India's population in 1974 . b. Substitute 27 for \(x\) and use your calculator to find India's population, to the nearest million, in the year 2001 as modeled by this function. c. Find India's popülation, to the nearest million, in the year 2028 as predicted by this function. d. Find India's population, to the nearest million, in the year 2055 as predicted by this function. e. What appears to be happening to India's population every 27 years?

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=\ln (x+1) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln \sqrt{x}+4=1 $$

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right)+2 \log _{4}(x+1) $$

The formula \(S-C(1+r)^{t}\) models inflation, where \(C\) - the value today, \(r-\) the annual inflation rate, and \(S\) - the inflated value \(t\) years from now. Use this formula to solve Exercises \(67-68\). Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 years?

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ h(x)=\ln (2 x) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{5} x+\log _{5}(4 x-1)=1 $$

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right] $$

The formula \(S-C(1+r)^{t}\) models inflation, where \(C\) - the value today, \(r-\) the annual inflation rate, and \(S\) - the inflated value \(t\) years from now. Use this formula to solve Exercises \(67-68\). Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 years?

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ h(x)=\ln \left(\frac{1}{2} x\right) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{6}(x+5)+\log _{6} x=2 $$

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right] $$

The formula \(S-C(1+r)^{t}\) models inflation, where \(C\) - the value today, \(r-\) the annual inflation rate, and \(S\) - the inflated value \(t\) years from now. Use this formula to solve Exercises \(67-68\). Round answers to the nearest dollar. If the inflation rate is \(3 \%,\) how much will a house now worth \(\$ 510,000\) be worth in 5 years?

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=2 \ln x $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{3}(x+6)+\log _{3}(x+4)=1 $$

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log x+\log \left(x^{2}-1\right)-\log 7-\log (x+1) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{6}(x+3)+\log _{6}(x+4)=1 $$

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log x+\log \left(x^{2}-4\right)-\log 15-\log (x+2) $$

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ h(x)=-\ln x $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{2}(x+2)-\log _{2}(x-5)=3 $$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

Use a calculator with an \(\overline{e^{x}}\) key to solve Exercises \(71-76\). Average annual premiums for employer-sponsored family health insurance policies more than doubled over II years. The bar graph shows the average cost of a family health insurance plan in the United States for six selected years from 2000 through 2011. The data can be modeled by $$ f(x)-782 x+6564 \text { and } g(x)-6875 e^{000 \pi x} $$ in which \(f(x)\) and \(g(x)\) represent the average cost of a family heallh insurance plan \(x\) years after 2000 . Use these functions to solve Exercises \(71-72\). Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of a family health insurance plan in \(2011 ?\) b. According to the exponential model, what was the average cost of a family health insurance plan in \(2011 ?\) c. Which function is a better model for the data in \(2011 ?\)

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ h(x)=\ln (-x) $$

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{4}(x+2)-\log _{4}(x-1)=1 $$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{6} 17 $$

Use a calculator with an \(\overline{e^{x}}\) key to solve Exercises \(71-76\). Average annual premiums for employer-sponsored family health insurance policies more than doubled over II years. The bar graph shows the average cost of a family health insurance plan in the United States for six selected years from 2000 through 2011. The data can be modeled by $$ f(x)-782 x+6564 \text { and } g(x)-6875 e^{000 \pi x} $$ in which \(f(x)\) and \(g(x)\) represent the average cost of a family heallh insurance plan \(x\) years after 2000 . Use these functions to solve Exercises \(71-72\). Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of a family health insurance plan in \(2008 ?\) b. According to the exponential model, what was the average cost of a family health insurance plan in \(2008 ?\) c. Which function is a better model for the data in \(2008 ?\)

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=2-\ln x $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used an exponential model with a positive growth rate to describe the depreciation in my car's value over four years.

Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 2 \log _{3}(x+4)=\log _{3} 9+2 $$

In college, we study large volumes of information information that, unfortunately, we do not often retain for very long. The function $$ f(x)-80 e^{-0.5 x}+20 $$ describes the percentage of information, \(f(x)\), that a particular person remembers \(x\) weeks after learning the information. a. Substitute 0 for \(x\) and, without using a calculator, find the percentage of information remembered at the moment it is first learned. b. Substitute 1 for \(x\) and find the percentage of information that is remembered after 1 week. c. Find the percentage of information that is remembered after 4 weeks d. Find the percentage of information that is remembered after one year ( 52 weeks).

The figure shows the graph of \(f(x)=\ln x\). Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=1-\ln x $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)