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Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve Exercises \(47-50\). A frozen steak initially has a temperature of \(28^{\circ} \mathrm{F}\). It is left to thaw in a room that has a temperature of \(75^{\circ} \mathrm{F}\). After 10 minutes, the temperature of the steak has risen to \(38^{\circ} \mathrm{F}\). After how many minutes will the temperature of the steak be \(50^{\circ} \mathrm{F} ?\)

begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$g(x)=\log _{2}(x+1)$$

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=100(4.6)^{x}$$

Graph \(y=3^{x}\) and \(x=3^{y}\) in the same rectangular coordinate system.

Describe a difference between exponential growth and logistic growth.

Use a calculator with a \({y^{x}}\) key or a \(\wedge\) key to solve India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential function \(f(x)=574(1.026)^{x}\) models the population of India, \(f(x),\) in millions, \(x\) years after 1974 a. Substitute 0 for \(x\) and, without using a calculator, find India's population in 1974 . b. Substitute 27 for \(x\) and use your calculator to find India's population, to the nearest million, in the year 2001 as modeled by this function. c. Find India's population, to the nearest million, in the year 2028 as predicted by this function. d. Find India's population, to the nearest million, in the year 2055 as predicted by this function. e. What appears to be happening to India's population every 27 years?

Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$2 \log _{3}(x+4)=\log _{3} 9+2$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used an exponential model with a positive growth rate to describe the depreciation in my car's value over four years.