Chapter 5

Compound Interest If \(\$ 600\) is invested at an interest rate of 2.5\(\%\) per year, find the amount of the investment at the end of 10 years for the following compounding methods. (a) Annually (b) Semiannully (c) Quarterly \(\quad\) (d) Continuously

Use the definition of the logarithmic function to find \(x\). $$ \text { (a) } \log _{x} 1000=3 \quad \text { (b) } \log _{x} 25=2 $$

Solve the equation. \(x^{2} 10^{x}-x 10^{x}=2\left(10^{x}\right)\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right) $$

Compound Interest If \(\$ 8000\) is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of 12 years for the following interest rates. \(\begin{array}{llll}{\text { (a) } 2 \%} & {\text { (b) } 3 \%} & {\text { (c) } 4.5 \%} & {\text { (d) } 7 \%}\end{array}\)

Use the definition of the logarithmic function to find \(x\). $$ \text { (a) } \log _{x} 16=4 \quad \text { (b) } \log _{x} 8=\frac{3}{2} $$

\(29-43\) . These exercises deal with logarithmic scales. Earthquake Magnitudes If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale?

Solve the equation. \(4 x^{3} e^{-3 x}-3 x^{4} e^{-3 x}=0\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}\left(\frac{x\left(x^{2}+1\right)}{\sqrt{x^{2}-1}}\right) $$

Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{x} 6=\frac{1}{2}} & {\text { (b) } \log _{x} 3=\frac{1}{3}}\end{array} $$

Solve the equation. \(X^{2} e^{X}+X e^{X}-e^{X}=0\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{5} \sqrt{\frac{x-1}{x+1}} $$

Compound Interest Which of the given interest rates and compounding periods would provide the better investment? (a) 5\(\%\) per year, compounded semiannully (b) 5\(\%\) per year, compounded continuously

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 2} & {\text { (b) } \log 35.2} & {\text { (c) } \log \left(\frac{2}{3}\right)}\end{array} $$

Solve the logarithmic equation for \(x.\) \(\ln x=10\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \left(x \sqrt{\frac{y}{z}}\right) $$

(a) Sketch the graphs of \(f(x)=2^{x}\) and \(g(x)=3\left(2^{x}\right)\) (b) How are the graphs related?

Investment A sum of \(\$ 5000\) is invested at an interest rate of 9\(\%\) per year, compounded continuously. (a) Find the value \(A(t)\) of the investment after \(t\) years. (b) Draw a graph of \(A(t) .\) (c) Use the graph of \(A(t)\) to determine when this investment will amount to \(\$ 25,000\) .

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 50} & {\text { (b) } \log \sqrt{2}} & {\text { (c) } \log (3 \sqrt{2})}\end{array} $$

Solve the logarithmic equation for \(x.\) \(\ln (2+x)=1\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \frac{3 x^{2}}{(x+1)^{10}} $$

(a) Sketch the graphs of \(f(x)=9^{x / 2}\) and \(g(x)=3^{x}\) (b) Use the Laws of Exponents to explain the relationship between these graphs.

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \ln 5} & {\text { (b) } \ln 25.3} & {\text { (c) } \ln (1+\sqrt{3})}\end{array} $$

\(29-43\) . These exercises deal with logarithmic scales. Earthquake Magnitudes The 1985 Mexico City earthquake had a magnitude of 8.1 on the Richter scale. The 1976 earthquake in Tangshan, China, was 1.26 times as intense. What was the magnitude of the Tangshan earthquake?

Solve the logarithmic equation for \(x.\) \(\log x=-2\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \sqrt[4]{x^{2}+y^{2}} $$

Compare the functions \(f(x)=x^{3}\) and \(g(x)=3^{x}\) by evaluating both of them for \(x=0,1,2,3,4,5,6,7,8,10,15,\) and \(20 .\) Then draw the graphs of \(f\) and \(g\) on the same set of axes.

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \ln 27} & {\text { (b) } \ln 7.39} & {\text { (c) } \ln 54.6}\end{array} $$

Solve the logarithmic equation for \(x.\) \(\log (x-4)=3\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x}{\sqrt[3]{1-x}}\right) $$

If \(f(x)=10^{x},\) show that \(\frac{f(x+h)-f(x)}{h}=10^{x}\left(\frac{10^{h}-1}{h}\right)\)

Sketch the graph of the function by plotting points. $$ f(x)=\log _{3} x $$

\(29-43\) . These exercises deal with logarithmic scales. Traffic Noise The intensity of the sound of traffic at a busy intersection was measured at \(2.0 \times 10^{-5} \mathrm{W} / \mathrm{m}^{2} .\) Find the intensity level in decibels

Solve the logarithmic equation for \(x.\) \(\log (3 x+5)=2\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{\frac{x^{2}+4}{\left(x^{2}+1\right)\left(x^{3}-7\right)^{2}}} $$

(a) Compare the rates of growth of the functions \(f(x)=2^{x}\) and \(g(x)=x^{5}\) by drawing the graphs of both functions in the following viewing rectangles. (i) \([0,5]\) by \([0,20]\) (ii) \([0,25]\) by \(\left[0,10^{7}\right]\) (iii) \([0,50]\) by \(\left[0,10^{8}\right]\) (b) Find the solutions of the equation \(2^{x}=x^{5}\) , rounded to one decimal place.

Sketch the graph of the function by plotting points. $$ g(x)=\log _{4} x $$

\(29-43\) . These exercises deal with logarithmic scales. Comparing Decibel Levels The noise from a power mower was measured at 106 \(\mathrm{dB}\) . The noise level at a rock concert was measured at 120 \(\mathrm{dB}\) . Find the ratio of the intensity of the rock music to that of the power mower.

Solve the logarithmic equation for \(x.\) \(\log _{3}(2-x)=3\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{x \sqrt{y} \sqrt{z}} $$

(a) Compare the rates of growth of the functions \(f(x)=3^{x}\) and \(g(x)=x^{4}\) by drawing the graphs of both functions in the following viewing rectangles: (i) \([-4,4]\) by \([0,20]\) (ii) \([0,10]\) by \([0,5000]\) (iii) \([0,20]\) by \(\left[0,10^{5}\right]\) (b) Find the solutions of the equation \(3^{x}=x^{4}\) , rounded to two decimal places.

Sketch the graph of the function by plotting points. $$ f(x)=2 \log x $$

\(29-43\) . These exercises deal with logarithmic scales. Inverse Square Law for Sound A law of physics states that the intensity of sound is inversely proportional to the square of the distance \(d\) from the source: \(I=k / d^{2} .\) (a) Use this model and the equation $$ B=10 \log \frac{I}{I_{0}} $$ (described in this section) to show that the decibel levels \(B_{1}\) and \(B_{2}\) at distances \(d_{1}\) and \(d_{2}\) from a sound source are related by the equation $$ B_{2}=B_{1}+20 \log \frac{d_{1}}{d_{2}} $$ (b) The intensity level at a rock concert is 120 \(\mathrm{dB}\) at a distance 2 \(\mathrm{m}\) from the speakers. Find the intensity level at a distance of \(10 \mathrm{m} .\)

Solve the logarithmic equation for \(x.\) \(4-\log (3-x)=3\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \left(\frac{x^{3} \sqrt{x-1}}{3 x+4}\right) $$

Draw graphs of the given family of functions for \(c=0.25,0.5,1,2,4 .\) How are the graphs related? $$ f(x)=c 2^{x} $$

Sketch the graph of the function by plotting points. $$ g(x)=1+\log x $$

Solve the logarithmic equation for \(x.\) \(\log _{2}\left(x^{2}-x-2\right)=2\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{10^{x}}{x\left(x^{2}+1\right)\left(x^{4}+2\right)}\right) $$

Draw graphs of the given family of functions for \(c=0.25,0.5,1,2,4 .\) How are the graphs related? $$ f(x)=2^{c x} $$