Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

Both the functions f and g are plotted on the same rectangular coordinate system. They intersect at x=1, and each function mirrors the other across the line y=x.

Step 1: 1. Plotting the Exponential Function

The exponential function \(f(x) = 5^{x}\) reflects the characteristic that the y-value increases rapidly with the increase of x. When x=0, y=1. By choosing a set of values for x and finding their corresponding y-values, we get a set of coordinates that can be plotted. Label the axes of the graph properly.

Step 2: 2. Plotting the Logarithmic Function

The function \(g(x) = \log_{5}{x}\) reflects that the y-value rises with the increase of x, but at a much slower rate than the exponential function. This function is not defined for x less than or equal to 0. By choosing a set of values for x and finding their corresponding y-values, we get a another set of coordinates that can be plotted on the same graph. Label it properly.

Step 3: 3. Analyze the Graphs

Now both functions are plotted on the same graph, it is visually clearly that they intersect at x=1. Also, we can observe that these functions mirror each other across the line y=x, which becomes apparent since the exponential function and the logarithmic function are inverses of each other.