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Chapter 4: Problem 4
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{x}=625$$
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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\)
Use the exponential growth model, \(A=A_{0} e^{k_{i}},\) to show that the time it takes a population to triple (to grow from \(A_{0}\) to \(\left.3 A_{0}\right)\) is given by \(t=\frac{\ln 3}{k}\)
Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. \(y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}\)
Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.
Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=4.3 e^{0.01 t}\) describes New Zealand's population, \(A,\) in millions, \(t\) years after 2010 . a. What is New Zealand's growth rate? b. How long will it take New Zealand to double its population?
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