91Ó°ÊÓ

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{3 x-1}=125$$

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbon-I4 present after t years. Use this model to solve Exercises \(15-16 .\) How many grams of carbon-14 will be present in \(11,430\) years?

The half-life of the radioactive element plutonium-239 is \(25,000\) years. If 16 grams of plutonium- 239 are initially present, how many grams are present after \(25,000\) years? \(50,000\) years? \(75,000\) years? \(100,000\) years? \(125,000\) years?

Use the exponential decay model for carbon- \(14, A=A_{0} e^{-0.000121 t}\) to solve Exercises \(19-20\) Skeletons were found at a construction site in San Francisco in \(1989 .\) The skeletons contained \(88 \%\) of the expected amount of carbon-14 found in a living person. In \(1989,\) how old were the skeletons?

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{5}\left(\frac{\sqrt{x}}{25}\right)\)

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(\left.2 A_{0}\right)\) is given by \(t=\frac{\ln 2}{k}\)

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 \(t=\frac{\ln 3}{k}\)

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 \(A=112.5 e^{0.012 y}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

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 affer 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.1 billion for \(2000 ?\)

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\log 250+\log 4\)