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Chapter 4: Problem 79
Find the domain of each logarithmic function. $$f(x)=\ln (x-2)^{2}$$
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Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function \(W(t)=2600\left(1-0.51 e^{-0.075 t}\right)^{3}\) models the weight, \(W(t),\) in kilograms, of a female African elephant at age \(t\) years. (1 kilogram \(=2.2\) pounds) Use a graphing utility to graph the function. Then \([\mathrm{TRACE}]\) along the curve to estimate the age of an adult female elephant weighing 1800 kilograms.
Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.
Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$
In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=100(4.6)^{x} $$
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