Problem 40 - College Algebra

Question

Graph \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

Plot the graphs \(f(x)=5^{x}\) that sharply increases for positive x-values, and \(g(x)=\log _{5} x\) that slowly increases for positive x-values.

Step 1: Identify the points

Choose some convenient values of x, substitute them into the function, and calculate the corresponding function values. For \(f(x)=5^{x}\), let's take x = -1, 0, and 1. For \(g(x)=\log _{5} x\), it's important to choose x-values that are powers of 5 because we can easily compute the logarithms

Step 2: Plot the points and draw the graph for \(f(x)=5^{x}\)

After calculating the corresponding function values, plot these points on the same rectangular coordinate grid and draw a smooth curve through the points. This graph is an exponential growth function, increasing rapidly for positive x-values

Step 3: Plot the points and draw the graph for \(g(x)=\log _{5} x\)

As in step 2, after computing the corresponding function values, plot these points on the same rectangular coordinate grid and draw a smooth curve through the points. This graph is the logarithmic function which is the inverse of the exponential growth function. It slowly increases for positive x-values but does not exist for negative x-values (since you can't take a logarithm of a negative number)