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Chapter 3: Problem 33
Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. $$f(x)=7^{x}, g(x)=\log _{7} x$$
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Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $g(x)=\log _{6} x$$
Sound Intensity The relationship between the number of decibels \(\beta\) and the intensity of a sound \(I\) in watts per square meter is \(\beta=10 \log \left(\frac{I}{10^{-12}}\right)\) (a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter. (b) Determine the number of decibels of a sound with an intensity of \(10^{-2}\) watt per square meter. (c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain.
Use the acidity model given by \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right],\) where acidity \((\mathbf{p H})\) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?
Home Mortgage \(A \$ 120,000\) home mortgage for 30 years at \(7 \frac{1}{2} \%\) has a monthly payment of \(\$ 839.06\) Part of the monthly payment covers the interest charge on the unpaid balance, and the remainder of the payment reduces the principal. The amount paid toward the interest is $$u=M-\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}$$ and the amount paid toward the reduction of the principal is $$v=\left(M-\frac{P r}{12}\right)\left(1+\frac{r}{12}\right)^{12 t}$$ In these formulas, \(P\) is the size of the mortgage, \(r\) is the interest rate, \(M\) is the monthly payment, and \(t\) is the time (in years). (a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 30 years of mortgage payments.) (b) In the early years of the mortgage, is the greater part of the monthly payment paid toward the interest or the principal? Approximate the time when the monthly payment is evenly divided between interest and principal reduction. (c) Repeat parts (a) and (b) for a repayment period of 20 years \((M=\$ 966.71) .\) What can you conclude?
Write the logarithmic equation in exponential form. $$\ln 7=1.945 \ldots$$
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