Use your calculator to estimate this value. Click "Submit" when finished.

You saw that the power of 2 that is 16 is 4, and the power of 2 that is 60 is approximately 5.907.

The first statement can also be expressed as "the logarithm base 2 of 16 is 4". Look to the right to see how this is written.

We say that the logarithm base 2 of a number is the power of 2 that equals that number. So, the logarithm base 2 of 60 is approximately 5.907.

You have seen that the logarithm base 2 of a number is the power of 2 that equals that number.

Use the graph and table of values for y = 2x to complete the table of values for y = log2x. Click "Submit" when finished.

How are the values in the table for y = 2x related to those in the table for y = log2x? What do you think the graph of y = log2x will look like?

Notice that the x- and y -values in the tables for y = 2x and y = log2x are reversed. So, y = 2x and y = log2x are inverses.

Let's look at the graph of y = log2x. Each point on the graph of y = log2x corresponds to a point on the graph of y = 2x. The coordinates of these points are reversed.

You have found the logarithms of different positive numbers. What about the logarithms of negative numbers? For example, what would the logarithm base 2 of -3 be? This is the same as finding the power of 2 that is -3.

Use the graphs, tables, and your calculator to answer this question. Click "Submit" when finished.

You can see on the graph of y = 2x that there are no points with negative y-values. So the graph of y = log2x does not contain any points with negative x-values. So, the logarithm of -3 is not defined.

Similarly, the logarithm of zero is also not defined. You can see on the graph of y = 2x that there is no point whose y-coordinate is 0. Therefore, there is no point on the graph of y = log2x whose x-coordinate is 0.

So far, you have looked at logarithms with base 2. We can also evaluate logarithms with other bases. For example, log39 is the power of 3 that is equal to 9. So, log39 = 2.

Similarly log1010,000 = 4 because 104=10,000, and logee3 = 3 because e3 = e3.

If the base of a logarithm is 10 it is called the common logarithm and we don't write the base. So, log1010000 is written log10000.

If the base of a logarithm is e, it is called the natural logarithm and is written as ln. So the logee3 is the natural log of e3, and it can be written ln e3.

Now let's look at the graphs of logarithms with different bases. Use the slider to change the value of b, the base of the logarithm. Notice how changing b affects the graph. You may check the box to look at the graph of the inverse exponential equation as well.

As you change the value of b, think about these questions: Does the base of the logarithm affect the x-intercept of the graph? Is it possible to find the logarithm, in any base, of a negative number? How does the shape of the graph change when the base is less than 1? How is the graph of a logarithm function related to the graph of its inverse exponential function? Click "Done Exploring" when finished.

The graph of y = logbx always crosses the x-axis at the point (1,0). Also, notice that the x-values are always greater than 0. Regardless of the base, it is not possible to find the logarithm of 0 or of a negative number. And, note that the graph of a logarithm function and its inverse exponential function are reflections across the line y = x.

The table and graph for the function y = 2x are displayed to the right.

Using the table or the graph, you can see that 16 can be written as 24. Can any number be written as a power of 2? For example, what power of 2 is 60?

To find this value, you can graph y = 2x and y = 60 and find their point of intersection.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education