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Chapter 3: Problem 27
Solve the exponential equation algebraically. Approximate the result to three decimal places. $$8\left(10^{3 x}\right)=12$$
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Graphical Analysis Use a graphing utility to graph \(f\) and \(g\) in the same viewing window and determine which is increasing at the greater rate as \(x\) approaches + \(\infty\). What can you conclude about the rate of growth of the natural logarithmic function? (a) \(f(x)=\ln x, \quad g(x)=\sqrt{x}\) (b) \(f(x)=\ln x, \quad g(x)=\sqrt[4]{x}\)
Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$g(x)=\ln (-x)$$
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?
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. The \(\mathrm{pH}\) of a solution decreases by one unit. By what factor does the hydrogen ion concentration increase?
Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $g(x)=\log _{6} x$$
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