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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 growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) By what percentage is the population of India increasing each year?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{1-5 x}=793$$

The logistic growth function $$ f(t)=\frac{500}{1+83.3 e^{-0.162 t}} $$ describes the population, \(f(t),\) of an endangered species of birds \(t\) years after they are introduced to a nonthreatening habitat. a. How many birds were initially introduced to the habitat? b. How many birds are expected in the habitat after 10 years? c. What is the limiting size of the bird population that the habitat will sustain?

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=4.5(0.6)^{x} $$

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

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{r t}\) to solve. Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 10,000\) for 5 years at an interest rate of \(5.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

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 \(I .\) Where possible, evaluate logarithmic expressions. $$ \ln x+\ln 7 $$

The exponential function \(f(x)=67.38(1.026)^{x}\) describes the population of Mexico, \(f(x),\) in millions, \(x\) years after 1980 a. Substitute 0 for \(x\) and, without using a calculator, find Mexico's population in 1980 . b. Substitute 27 for \(x\) and use your calculator to find Mexico's population in the year 2007 as predicted by this function. c. Find Mexico's population in the year 2034 as predicted by this function. d. Find Mexico's population in the year 2061 as predicted by this function. e. What appears to be happening to Mexico's population every 27 years?

The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere.The function \(f(x)=1000(0.5)^{x / 30}\) describes the amount, \(f(x),\) in kilograms, of cesium-137 remaining in Chernobyl \(x\) years after \(1986 .\) If even 100 kilograms of cesium-137 remain in Chernobyl's atmosphere, the area is considered unsafe for human habitation. Find \(f(80)\) and determine if Chernobyl will be safe for human habitation by 2066