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Chapter 3: Problem 34

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. $$f(x)=5^{x}, g(x)=\log _{5} x$$

Short Answer

Expert verified
The graphs of functions \(f(x) = 5^{x}\) and \(g(x) = \log_{5} x\) in the same coordinate plane are sketched. The important thing to remember is that the graph of an exponential function is increasing rapidly and is always above the x-axis, while the graph of a logarithmic function starts from negative infinity at \(x=1\) and increases very slowly.

Step by step solution

01

Understanding the exponential function

First, let's look at the exponential function \(f(x) = 5^{x}\). One property of exponential functions is that for any positive number \(x\), the function will also be positive because any number raised to a power is positive. The graph of \(f(x) = 5^{x}\) starts near the origin (0,0) and increases rapidly as \(x\) becomes larger. As x approaches negative infinity, the graph approaches the x-axis but never touches or crosses it, because, theoretically, a number with a positive base never becomes zero.
02

Understanding the logarithmic function

Now let's look at the logarithmic function \(g(x) = \log_{5} x\). This is the inverse function of \(f(x) = 5^{x}\). Recall that the logarithmic function starts at negative infinity when \(x=1\) and increases slowly. As \(x\) moves towards positive infinity, the function increases indefinitely but at a slower and slower rate. It crosses the x-axis at \(x = 1\). It is undefined at \(x = 0\) and \(x < 0\) since log of zero and negative numbers are undefined.
03

Sketching the graphs

Start by sketching \(f(x) = 5^{x}\), which starts from the point (0,1), crosses the y-axis at (0,1) and keeps increasing steeply as \(x\) moves positively. On the negative side of \(x\), the graph gets closer and closer to the x-axis but never touches or crosses it. Next, sketch \(g(x) = \log_{5} x\). It crosses the x-axis at the point (1,0) and increases slowly as \(x\) becomes larger. On the left of the y-axis from x=0 to negative side, the function is undefined.

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