Problem 44 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The graphs of \(f(x) = 5^x\) and \(g(x) = \log_5{x}\) are plotted together. Both functions exhibit different behaviors. The function \(5^x\) grows rapidly while \(\log_5{x}\) grows slowly. Also, \(5^x\) approaches zero as x decrease while \(\log_5{x}\) approaches negative infinity as x approaches zero from the positive side.

Step 1: Graphing the Exponential Function

The function \(f(x) = 5^x\) is an exponential function. The graph of this function will pass through the point (0,1) since any number raised to the power of 0 is 1. It will also pass through the point (1,5) since 5 to the power of 1 is 5. The graph will grow rapidly as x increase and will approach zero as x decrease. Plot these points and sketch the curve based on them.

Step 2: Graphing the Logarithmic Function

The function \(g(x) = \log_5{x}\) is a logarithmic function. One key point is given by (1,0) since the logarithm of 1 in any base is 0. The second point is (5,1) since the logarithm of the base itself is always equal to 1. The graph will increase slowly as x increases and will approach negative infinity as x approach zero from the positive side. Plot these points and draw the logarithmic curve.

Step 3: Bringing Both Graphs Together

Now that we have the individual graphs, they can be put up on the same graph. This will present the overview of both functions together. This is the final model demonstrating the exponential and logarithmic behavior on the same graph.