Chapter 5

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

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

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 \(x\) $$ \log (x-4)=3 $$

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

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

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

Use the Laws of Logarithms to combine the expression. $$ \log _{3} 5+5 \log _{3} 2 $$

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 \(x\) $$ \log _{3}(2-x)=3 $$

Use the Laws of Logarithms to combine the expression. $$ \log 12+\frac{1}{2} \log 7-\log 2 $$

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 \(B_{B}\) at distances \(d\) 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 \(x\) $$ 2-\ln (3-x)=0 $$

Use the Laws of Logarithms to combine the expression. $$ \log _{2} A+\log _{2} B-2 \log _{2} C $$

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

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

Use the Laws of Logarithms to combine the expression. $$ \log _{5}\left(x^{2}-1\right)-\log _{5}(x-1) $$

(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.

Solve the logarithmic equation for \(x\) $$ \log _{2} 3+\log _{2} x=\log _{2} 5+\log _{2}(x-2) $$

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

Solve the logarithmic equation for \(x\) $$ 2 \log x=\log 2+\log (3 x-4) $$

Use the Laws of Logarithms to combine the expression. $$ \ln (a+b)+\ln (a-b)-2 \ln c $$

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

Solve the logarithmic equation for \(x\) $$ \log x+\log (x-1)=\log (4 x) $$

Use the Laws of Logarithms to combine the expression. $$ \ln 5+2 \ln x+3 \ln \left(x^{2}+5\right) $$

Solve the logarithmic equation for \(x\) $$ \log _{5} x+\log _{5}(x+1)=\log _{5} 20 $$

Use the Laws of Logarithms to combine the expression. $$ 2\left(\log _{5} x+2 \log _{5} y-3 \log _{5} z\right) $$

Solve the logarithmic equation for \(x\) $$ \log _{5}(x+1)-\log _{5}(x-1)=2 $$

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

Solve the logarithmic equation for \(x\) $$ \log x+\log (x-3)=1 $$

Use the Laws of Logarithms to combine the expression. $$ \log _{a} b+c \log _{a} d-r \log _{a} s $$

Draw the graph of \(y=3^{x},\) then use it to draw the graph of \(y=\log _{3} x .\)

Solve the logarithmic equation for \(x\) $$ \log _{9}(x-5)+\log _{9}(x+3)=1 $$

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{2} 5 $$

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

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{5} 2 $$

For what value of \(x\) is the following true? $$ \log (x+3)=\log x+\log 3 $$

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{3} 16 $$

(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 \(2^{x}=x^{5},\) correct to one decimal place.

For what value of \(x\) is it true that \((\log x)^{3}=3 \log x ?\)

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{6} 92 $$

(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 \(3^{x}=x^{4},\) correct to two decimal places.

Solve for \(x : \quad 2^{2 / \log _{x} x}=\frac{1}{16}\)

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{7} 2.61 $$

53–54 ? Draw graphs of the given family of functions for c = 0.25, 0.5, 1, 2, 4. How are the graphs related? $$ f(x)=c 2^{x} $$

Solve for \(x : \quad \log _{2}\left(\log _{3} x\right)=4\)

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{6} 532 $$

53–54 ? Draw graphs of the given family of functions for c = 0.25, 0.5, 1, 2, 4. How are the graphs related? $$ f(x)=2^{c x} $$

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ \ln x=3-x $$

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{4} 125 $$