Problem 43 - College Algebra

Question

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graphs of \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) intersect at the point (1, 1). The graph of the function \(f(x)=4^{x}\) starts from (0,1) and increases rapidly as x increases. The graph of the function \(g(x)=\log_4{x}\) starts from negative infinity at x=0 and rises slowly as x increases.

Step 1: Understand the Functions

First, understand our two functions. Function \(f(x) = 4^x\) is an exponential function, where the base is 4. The graph of \(f(x) = 4^x\) is a curve that starts at (0,1) when x is 0 and increases rapidly as x increases. Function \(g(x) = log_4(x)\) is a logarithmic function, where the base is 4. Its graph starts from negative infinity at x = 0 and slowly rises as x increases. It crosses the x-axis at the point (1,0).

Step 2: Plot the Graphs

Next, graph both functions in the same Cartesian plane. Start by graphing \(y = 4^x\). This is the graph of an exponential function which grows rapidly as x gets larger. Then graph \(y = log_4(x)\). This is the graph of a logarithmic function, which grows much slower than the exponential function.

Step 3: Identify the Intersection Point

Then identify the point where the two graphs intersect. As base of exponential and logarithmic function is the same, they intersect at point (1,1). So plot this point on our graph and label it.

Step 4: Final Check

Finally, check your work. Ensure that each graph has been plotted correctly. The exponential graph should grow rapidly, starting from the point (0, 1) and the logarithmic graph should rise slowly, starting from negative infinity at x = 0. And both graphs should intersect at the point (1,1).