Chapter 12

Solve. Use \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Round answers to two decimal places. Find the amount accrued if \(\$ 500\) is invested and earns \(7 \%\) compounded monthly for 4 years.

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{2} y^{3} z $$

Psychologists call the graph of the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\) the learning curve since the formula relates time t passed, in weeks, to a measure \(N\) of learning achieved, to a measure \(A\) of maximum learning possible, and to a measure c of an individual's learning style. Use this formula to answer. Round to the nearest whole number. Janine Jenkins is working on her dictation skills. She wants to take dictation at a rate of 150 words per minute and believes that the maximum rate she can hope for is 210 words per minute. How many weeks should it take her to achieve the 150 -word level if \(c\) is \(0.07 ?\)

Solve. $$ \log _{x} 49=2 $$

Use the formula \(A=P e^{r t}\) to solve. How much money does Barbara Mack owe at the end of 4 years if \(6 \%\) interest is compounded continuously on her \(\$ 2000\) debt?

If \(f(x)=3^{x}\), find each value. give an exact answer and a two-decimal-place approximation. See Sections 8.2 and \(10.2 .\) $$ f(2) $$

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{5} x^{3}(x+1) $$

Psychologists call the graph of the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\) the learning curve since the formula relates time t passed, in weeks, to a measure \(N\) of learning achieved, to a measure \(A\) of maximum learning possible, and to a measure c of an individual's learning style. Use this formula to answer. Round to the nearest whole number. A psychologist is measuring human capability to memorize nonsense syllables. How many weeks should it take a subject to learn 15 nonsense syllables if the maximum possible to learn is 24 syllables and \(c\) is \(0.17 ?\)

Solve. $$ \log _{x} 8=3 $$

Use the formula \(A=P e^{r t}\) to solve. Find the amount of money for which a \(\$ 2500\) certificate of deposit is redeemable if it has been earning \(10 \%\) interest compounded continuously for 3 years.

If \(f(x)=3^{x}\), find each value. give an exact answer and a two-decimal-place approximation. See Sections 8.2 and \(10.2 .\) $$ f(0) $$

Solve each equation. $$ 3 x-7=11 $$

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{x^{2}-y+2 z}{3 x} $$

Solve. $$ \log _{2} \frac{1}{8}=x $$

Approximate each logarithm to four decimal places. $$ \log _{2} 3 $$

If \(f(x)=3^{x}\), find each value. give an exact answer and a two-decimal-place approximation. See Sections 8.2 and \(10.2 .\) $$ f\left(\frac{1}{2}\right) $$

Solve each equation. $$ \begin{aligned} &3 x-4=3(x+1)\\\ \end{aligned} $$

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{6} \frac{x^{2}}{x+3} $$

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{x^{3}-2 y+z}{2 z} $$

Solve. $$ \log _{3} \frac{1}{81}=x $$

Approximate each logarithm to four decimal places. $$ \log _{3} 2 $$

If \(f(x)=3^{x}\), find each value. give an exact answer and a two-decimal-place approximation. See Sections 8.2 and \(10.2 .\) $$ f\left(\frac{1}{2}\right) $$

Solve each equation. $$ 2-6 x=6(1-x) $$

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{3} \frac{(x+5)^{2}}{x} $$

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{3 z-4 x+y}{x+2 z} $$

Solve. $$ \log _{3} \frac{1}{27}=x $$

Approximate each logarithm to four decimal places. $$ \log _{8} 6 $$

Solve. See the Concept Check in this section. Suppose that \(f\) is a one-to-one function and that \(f(2)=9\) a. Write the corresponding ordered pair. b. Name one ordered pair that we know is a solution of the inverse of \(f\), or \(f^{-1}\).

Is the given function an exponential function? $$ f(x)=1.5 x^{2} $$

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \frac{5}{3} $$

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{4 y-3 x+z}{2 x+y} $$

Solve. $$ \log _{5} \frac{1}{125}=x $$

Approximate each logarithm to four decimal places. $$ \log _{6} 8 $$

Solve. See the Concept Check in this section. Suppose that \(F\) is a one-to-one function and that \(F\left(\frac{1}{2}\right)=-0.7\) a. Write the corresponding ordered pair. b. Name one ordered pair that we know is a solutic of the inverse of \(F,\) or \(F^{-1}\).

Is the given function an exponential function? $$ g(x)=3^{x} $$

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} 25 $$

The formula \(y=y_{0} e^{k t}\) gives the population size y of a population that experiences a relative growth rate \(k(k\) is positive if growth is increasing and \(k\) is negative if growth is decreasing). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time \(0 .\) Use this formula to solve Exercises 55 and \(56 .\) Round answers to the nearest year. (Source for data: U.S. Census Bureau and Federal Reserve Bank of Chicago) In \(2009,\) the population of Michigan was approximately 9,970,000 and decreasing according to the formula \(y=y_{0} e^{-0.003 t}\). Assume that the population continues to decrease according to the given formula and predict how many years after which the population of Michigan will be \(9,500,000 .\) (Hint: Let \(y_{0}=9,970,000 ; y=9,500,000\), and solve for \(t\).)

Solve. $$ \log _{8} x=\frac{1}{3} $$

Approximate each logarithm to four decimal places. $$ \log _{4} 9 $$

Is the given function an exponential function? $$ h(x)=\left(\frac{1}{2} x\right)^{2} $$

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} 15 $$

The formula \(y=y_{0} e^{k t}\) gives the population size y of a population that experiences a relative growth rate \(k(k\) is positive if growth is increasing and \(k\) is negative if growth is decreasing). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time \(0 .\) Use this formula to solve. Round answers to the nearest year. (Source for data: U.S. Census Bureau and Federal Reserve Bank of Chicago) In \(2009,\) the population of Illinois was approximately 12,910,000 and increasing according to the formula \(y=y_{0} e^{0.005 t}\). Assume that the population continues to increase according to the given formula and predict how many years after which the population of Illinois will be 13,500,000 . (See the Hint for Exercise \(55 .\) )

Solve. $$ \log _{9} x=\frac{1}{2} $$

Approximate each logarithm to four decimal places. $$ \log _{9} 4 $$

Is the given function an exponential function? $$ F(x)=0.4^{x+1} $$

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \frac{3}{5} $$

When solving a logarithmic equation, explain why you must check possible solutions in the original equation.

Solve. $$ \log _{4} 16=x $$

Approximate each logarithm to four decimal places. $$ \log _{3} \frac{1}{6} $$

If you are given the graph of a function, describe how you can tell from the graph whether the function has an inverse.