Problem 124 - College Algebra

Question

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the graph of \(g(x)=\log _{2} x\).

Step-by-Step Solution

Answer

To obtain the graph of \(g(x)=\log_{2} (x)\) from the graph of \(f(x)=2^{x}\), one must reflect the graph of \(f(x)\) with respect to the line \(y = x\) (since they are inverse functions). This gives a graph starting at (1,0) and increasing slowly as \(x\) increases. The y-axis serves as an asymptote, and the function is undefined for \(x\) less than or equal to 0.

Step 1: Overview of the function \(f(x)=2^{x}\)

Initially, let's consider the function \(f(x)=2^{x}\). This is an exponential function, it starts at the point (0,1) for \(x=0\) and increases rapidly as \(x\) increases. The curve never touches the x-axis, thus the x-axis is an asymptote to the function. It's graph will be useful as we transform it into the logarithmic function.

Step 2: Understanding the logarithmic function

Now let's consider the function \(g(x)=\log_{2}(x)\). For exponential functions like \(f(x)=2^{x}\), the base is 2 and \(x\) is the exponent. Logarithmic functions are the inverse of exponential functions, which means for \(g(x)=\log_{2}(x)\), 2 is still the base, but \(x\) is now the number we are taking the log of, not the exponent. This means that \(g(x)\) asks the question, what power do we have to raise 2 to in order to get \(x\).

Step 3: Plotting \(g(x)=\log_{2} (x)\)

To get the graph of \(g(x)=\log_{2}(x)\) from \(f(x)=2^{x}\), reflect the graph of \(f(x)\) with respect to the line \(y = x\). This is because logarithmic functions are the inverse of exponential functions. This new graph will start at (1,0) because the logarithm of 1 in any base is 0. The curve approaches the y-axis but never crosses it, so the y-axis is an asymptotical line to the graph of this function. Additionally, the function is not defined for values of \(x\) less than or equal to 0.