Graphs of Logarithmic Functions and Exponential Functions

In response to a question about what these graphs look like, I used Fathom software to create the graphs of

Since Fathom, like your graphing calculator, has no log-base-2 function built in, I needed to use a law of logarithms to enter it another way... I used the change of base rule to change it to log of x divided by log of 2. Click here to see the graph that resulted.

In this previous graph, hopefully you'll notice the relationship between the two graphs. The log graph has an asymptote along the negative y-axis, passes through (1, 0) and continues to rise (slower and slower) after that. The exponential graph has an asymptote along the negative x-axis, passes through (0, 1) and continues to rise (faster and faster) after that. The two graphs are reflections of each other in the y=x line; this occurs because the functions are inverses of each other.

Next, I added to this graph the functions

With these additions, the graph has four different curves on it. Click here to see the graph now.

Notice that the log-base-3 curve has the same basic shape as log-base-2, except that it's closer to the asymptote and levels off more quickly. It passes through the same point on the x-axis as log-base-2: the point (1, 0).

Notice that the 3-to-the-x curve has the same basic shape as the 2-to-the-x curve except that it's steeper. It passes through the y-axis at the same point, namely (0, 1).

You can use a graphing calculator to try graphing other logarithmic or exponential functions. You can also read more about these functions at many different web pages, including this one.