Chapter 4

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{10} x=2\) (b) \(\log _{5} x=2\)

These exercises deal with logarithmic scales. The hydrogen ion concentrations in cheeses range from \(4.0 \times 10^{-7} \mathrm{M}\) to \(1.6 \times 10^{-5} \mathrm{M}\). Find the corresponding range of pH readings.

Solve the equation. $$x^{2} 2^{x}-2^{x}=0$$

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) Semiannually (c) Quarterly (d) Continuously

Use the Laws of Logarithms to expand the expression. $$\log \left(\frac{x^{3} y^{4}}{z^{6}}\right)$$

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

These exercises deal with logarithmic scales. The pH readings for wines vary from 2.8 to 3.8. Find the corresponding range of hydrogen ion concentrations.

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

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

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

These exercises deal with logarithmic scales. If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale?

Compound Interest Which of the given interest rates and compounding periods would provide the best investment? (a) \(2 \frac{1}{2} \%\) per year, compounded semiannually (b) \(2 \frac{1}{4} \%\) per year, compounded monthly (c) \(2 \%\) per year, compounded continuously

Solve the equation. $$4 x^{3} e^{-3 x}-3 x^{4} e^{-3 x}=0$$

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 \(x .\) (a) \(\log _{x} 6=\frac{1}{2}\) (b) \(\log _{x} 3=\frac{1}{3}\)

These exercises deal with logarithmic scales. The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time in Japan an earthquake with magnitude 4.9 caused only minor damage. How many times more intense was the San Francisco earthquake than the Japanese earthquake?

Compound Interest Which of the given interest rates and compounding periods would provide the better investment? (a) \(5 \frac{1}{4} \%\) per year, compounded semiannually (b) \(5 \%\) per year, compounded continuously

Solve the equation. $$x^{2} e^{x}+x e^{x}-e^{x}=0$$

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

Use a calculator to evaluate the expression, correct to four decimal places. (a) \(\log 2\) (b) \(\log 35.2\) (c) \(\log \left(\frac{2}{3}\right)\)

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 \(\$ 25,000.\)

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

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

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

Use a calculator to evaluate the expression, correct to four decimal places. (a) \(\log 50\) (b) \(\log \sqrt{2}\) (c) \(\log (3 \sqrt{2})\)

The Definition of \(e\) Illustrate the definition of the number \(e\) by graphing the curve \(y=(1+1 / x)^{x}\) and the line \(y=e\) on the same screen, using the viewing rectangle \([0,40]\) by \([0,4].\)

These exercises deal with logarithmic scales. The Northridge, California, earthquake of 1994 had a magnitude of 6.8 on the Richter scale. A year later, a 7.2-magnitude earthquake struck Kobe, Japan. How many times more intense was the Kobe earthquake than the Northridge earthquake?

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

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

Use a calculator to evaluate the expression, correct to four decimal places. (a) \(\ln 5\) (b) \(\ln 25.3\) (c) \(\ln (1+\sqrt{3})\)

These exercises deal with logarithmic scales. 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 \(x .\) $$\log x=-2$$

Compare the functions \(f(x)=x^{3}\) and \(g(x)=3^{x}\) by evaluating both of them for \(x=0,1,2,3,4,5,6,7,8,9,10,15,\) and 20 Then draw the graphs of \(f\) and \(g\) on the same set of axes.

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

Use a calculator to evaluate the expression, correct to four decimal places. (a) \(\ln 27\) (b) \(\ln 7.39\) (c) \(\ln 54.6\)

These exercises deal with logarithmic scales. The intensity of the sound of a subway train was measured at 98 dB. Find the intensity in \(\mathrm{W} / \mathrm{m}^{2}\).

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

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

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

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

These exercises deal with logarithmic scales. 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 \(x .\) $$\log (3 x+5)=2$$

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

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

These exercises deal with logarithmic scales. The noise from a power mower was measured at 106 dB. The noise level at a rock concert was measured at 120 dB. Find the ratio of the intensity of the rock music to that of the power mower.

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

(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.

Use the Laws of Logarithms to expand the expression. $$\log \sqrt{x \sqrt{y \sqrt{z}}}$$

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 \(10 \mathrm{m}\).