This interactive graph is designed to demonstrate the core relationship between exponential functions ($$y = b^x$$) and logarithmic functions ($$y = \log_b(x)$$).
The most important takeaway is that the blue curve (exponential function $$y = b^x$$) and the red curve (logarithmic function $$y = \log_b(x)$$) are inverse functions of each other. This means they "undo" each other's operation. For example, $$\log_b(b^x) = x$$.
Geometrically, the graphs of inverse functions are always symmetrical with respect to the line $$y = x$$. The gray dashed line in the graph represents $$y = x$$, and you can see how the blue and red curves mirror each other across this line.
The slider allows you to change the value of the base `b`. Observe what happens as you adjust it:
This illustrates the "compression effect" of logarithms, which allows them to effectively handle and represent values that span a vast range, such as in the Richter scale for earthquakes or decibels for sound intensity.
Another key relationship between these functions is in their domain and range:
This is a fundamental property of inverse functions: the domain of one is the range of the other, and vice versa.
Using the graph and your understanding of inverse functions, if the exponential function passes through the point \( (2, 9) \), what point must the logarithmic function pass through?
The correct answer is A. Since exponential and logarithmic functions are inverses, they reverse the x and y coordinates. If the exponential function has the point (2, 9), the inverse logarithmic function must have the point (9, 2).