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Chapter 4: Problem 64

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

Short Answer

Expert verified
You would likely prefer your salary to be modeled exponentially, because exponential growth results in larger increases over time compared to logarithmic growth, meaning you would earn more money over the same period.

Step by step solution

01

Understand exponential and logarithmic functions

An exponential function is a mathematical function of the form \(f(x) = a \cdot b^{x}\), where \(a\) and \(b\) are constants, \(a\) 鈮� 0, \(b\) > 0 , \(b\) 鈮� 1, and \(x\) is any real number. If your salary is modeled exponentially, it means your salary would increase at a consistently increasing rate. A logarithmic function is of the form \(f(x) = a + b \cdot log_{n}(x)\), where \(a\), \(b\), and \(n\) are constants and \(n\) > 0, \(n\) 鈮� 1. If your salary is modeled logarithmically, it means your salary would increase, but the rate of increase would slow down as your salary grows.
02

Compare exponential and logarithmic growth

The exponential function grows much faster than the logarithmic function. That means, if your salary were to be modeled exponentially, it would increase at a faster rate over the same period, as compared to if it were modeled logarithmically.
03

Understand the implications for salary

If you could choose, you would probably prefer your salary to grow exponentially. That's because exponential growth can result in much larger increases over time. On the other hand, logarithmic growth would result in smaller increases the larger the salary gets, which means the growth of your salary would slow down over time.

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Most popular questions from this chapter

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ 5^{x}=3 x+4 $$

Find the domain of each logarithmic function. $$ f(x)=\ln (x-2)^{2} $$

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.
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