Saying that *log _{a}M
=x *means exactly the same thing as saying

In other words:

Keep this in mind in thinking about logarithms. It makes lots of things obvious.

For example: What is *log _{2}
8? Ask yourself "To what power should I raise 2 in order to get
8?" *Since 8 is 2

Try to guess the general rule that these examples suggest. Then click the button below. Be sure to close the little windows that appears after you have looked at it.

Here's another way that remembering the rule:

can make some
things almost obvious. For example, what is *2
^{log}*

But * ^{log}_{2}^{
5 }* is the number to which you raise 2 in order to get 5!

*2 ^{log}2^{
5 }= 5. *Try the
following to check your understanding.

Try to guess the general rule that these examples suggest. Then click the button below. (Close the little window after you've looked at it.)

Let's use

to understand logarithms of product. For example: What is* log _{2}(8*32)*?

Notice that *8=2 ^{3
}*and 32

But this means that

*log _{2}(8*32)=log_{2}(2^{8})
= 8 = 3+5=*

*log _{2}(2^{3})+log_{2}(2^{5})=log_{2}(8)+log_{2}(32)*

*In
other words, the l og of the product 8*32 equals the sum
of the logs of 8 and 32.*

Of course there is nothing special about the base 2. The same idea holds for other logarithms.

Apply this idea to the following examples:

So what is the general rule?

*Precalculus Tutorials, B. Kaskosz and L. Pakula, 2002.*