Saying   that    log_aM =x  means exactly the same thing as saying a^x = M .

In other words:

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

For example:  What is log₂ 8?  Ask yourself "To what power should I raise 2 in order to get 8?"  Since 8 is 2³  the answer is "3."  So  log₂ 8=3.   Answer the following questions this way.

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 ^log2 ⁵?   Note that ^log₂ ⁵  is the power to which  2 is being raised. 

 But ^log₂ ⁵  is the number to which you raise 2 in order to get 5!  So if you raise 2 to that number you get 5!!  In other words 

2 ^log2 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 log₂(8*32)?  

Notice that 8=2³ and 32=2⁵  so 8*32=2³2⁵  = 2³⁺⁵ =2⁸ .  

But this means  that 

log₂(8*32)=log₂(2⁸) = 8 = 3+5=

log₂(2³)+log₂(2⁵)=log₂(8)+log₂(32)

In other words, the log 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.

So what is the general rule?

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