Problem 44 - Algebra and Trigonometry

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 the function \(f(x) = 5^x\) is a curve above the x-axis starting at point (0,1) and increasing steeply towards the right. The graph of function \(g(x) = log_5 x\) is a curve above the x-axis starting at point (1,0) and increasing gently towards the right. The two graphs intersect at points (1,1) and (0,0).

Step 1: Identify the Function Type

Identify the two functions as exponential and logarithmic functions. The function \(f(x) = 5^x\) is an exponential function with a base of 5 and \(g(x) = \log_5 x\) is a logarithmic function with a base of 5. Exponential functions have a characteristic 'J' shape, whereas logarithmic functions are best described as having a characteristic 'C' shape.

Step 2: Plot the Exponential Function

Plot the function \(f(x) = 5^x\). For an exponential function, the 'y'-intercept is always at (0,1). As 'x' increases, 'y' increases sharply, and as 'x' decreases, 'y' approaches zero but never reaches it.

Step 3: Plot the Logarithmic Function

Plot the function \(g(x) = log_5 x\). For a logarithmic function, the 'x'-intercept is always at (1,0). As 'x' increases, 'y' increases slowly, and as 'x' decreases towards zero, 'y' approaches negative infinity.

Step 4: Convergence of the Functions

On observing the plots, it can be noticed that the two functions intersect at points (1,1) and (0,0). This is because for all positive 'a', we have \(\log_a a = 1\) and \(a^0 = 1\). Thus, the two functions are reflections of each other across the line 'y = x'.