91Ó°ÊÓ

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) When will world population reach 7 billion?

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) When will world population reach 8 billion?

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. $$ \log 5+\log 2 $$

Evaluate each expression without using a calculator. $$ 8^{\log _{8} 19} $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) According to the model, what is the limiting size of the population that Earth will eventually sustain?

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. $$ \log 250+\log 4 $$

Evaluate each expression without using a calculator. $$7^{\log _{7} 23}$$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{2 x+1}-3^{x+2} $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 20 -year-olds have some coronary heart disease?

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.