Problem 44 - 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

The graph of \(f(x) = 5^x\) will be a curve increasing rapidly for positive x-values and approaching \(y=0\) for negative x-values. The graph of \(g(x) = \log_{5} x\) will be a curve increasing at a decreasing rate for x-values larger than 0. The two functions intersect at the point (1,5).

Step 1: Graph the exponential function

Start by plotting \(f(x) = 5^x\), the exponential function. Even for negative x-values, the function will always be positive. For \(x=0\), \(y=1\) because any number to the power of 0 is 1. For \(x=1\), \(y=5\) because 5 to the power 1 equals 5. Therefore, the graph will increase rapidly for positive x-values and approach \(y=0\) for negative x-values, never reaching \(y=0\).

Step 2: Graph the logarithmic function

Next, plot \(g(x) = \log_{5} x\), the logarithmic function. For this function, only positive x-values are allowed, as the logarithm of a negative number is not a real number. And the function equals 0 at \(x=1\) because the base 5 raised to the power 0 gives 1. The logarithm of 5 to the base 5 equals 1, because 5 to the power 1 equals 5. So like any logarithmic function, this graph will increase at a decreasing rate.

Step 3: Identify intersection points

Finally, identify where the two graphs intersect. It should be at \(x=1, y=5\). The reason for this intersection is that exponential and logarithmic functions are inverse of each other.