5.7E: Exponential and Logarithmic Equations (Exercises)
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38. Solve \(216^{3 x} \cdot 216^{x}=36^{3 x+2}\) by rewriting each side with a common base.
39. Solve \(\frac{125}{\left(\frac{1}{625}\right)^{-x-3}}=5^{3}\) by rewriting each side with a common base.
40. Use logarithms to find the exact solution for \(7 \cdot 17^{-9 x}-7=49\). If there is no solution, write no solution.
41. Use logarithms to find the exact solution for \(3 e^{6 n-2}+1=-60\). If there is no solution, write no solution.
42. Find the exact solution for \(5 e^{3 x}-4=6\). If there is no solution, write no solution.
43. Find the exact solution for \(2 e^{5 x-2}-9=-56\). If there is no solution, write no solution.
44. Find the exact solution for \(5^{2 x-3}=7^{x+1}\). If there is no solution, write no solution.
45. Find the exact solution for \(e^{2 x}-e^{x}-110=0\). If there is no solution, write no solution.
46. Use the definition of a logarithm to solve. \(-5 \log _{7}(10 n)=5\).
47. Use the definition of a logarithm to find the exact solution for \(9+6 \ln (a+3)=33\).
48. Use the one-to-one property of logarithms to find an exact solution for \(\log _{8}(7)+\log _{8}(-4 x)=\log _{8}(5)\). If there is no solution, write \(n o\) solution.
49. Use the one-to-one property of logarithms to find an exact solution for \(\ln (5)+\ln \left(5 x^{2}-5\right)=\ln (56)\). If there is no solution, write no solution.
50. The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10 \log \left(\frac{I}{I_{0}}\right),\) where \(I\) is the intensity of the sound in watts per square meter and \(I_{0}=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of \(6.3 \cdot 10^{-3}\) watts per square meter?
51. The population of a city is modeled by the equation \(P(t)=256,114 e^{0.25 t}\) where \(t\) is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?
52. Find the inverse function \(f^{-1}\) for the exponential function \(f(x)=2 \cdot e^{x+1}-5\).
53. Find the inverse function \(f^{-1}\) for the logarithmic function \(f(x)=0.25 \cdot \log _{2}\left(x^{3}+1\right)\).