Definition and Basics of Logarithms--Part II

Saying   that    logaM =x  means exactly the same thing as saying ax = M .

In other words:

logais the number to which you raise a in order to get M.

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 2the answer is "3."  So  log2 8=3.   Answer the following questions this way.

log3 9=

log2 (25)=    Hint:  To what power should I raise 2  to get 25 ? It's OBVIOUS!

log4 (47)=    Hint:  Use the idea of the previous problem!

loga (a7)=    Hint:  Look at the pattern in the previous two problems.

loga (a3)=  Hint:  Can you guess the general rule?

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:

"logais the number to which you raise a in order to get M."

can make some things almost obvious.  For example, what is  2 log2 5?   Note that log2is the power to which  2 is being raised.

But log2 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.

3 log3 7 Hint:  To what power do you raise 3 to get 7?  Remember what log3 means.

2 log2 7

a loga 7 Hint:  Can you guess the general rule?

a loga 3 Hint:  Can you guess the general rule?

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

"logais the number to which you raise a in order to get M,"

to understand logarithms of product.  For example:  What is log2(8*32)?

Notice that 8=23 and 32=2so 8*32=232 = 23+5 =2.

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.

Apply this idea to the following examples:

If log264=7 and log2256=8  then log2(64*256) =

If log3x=11 and log3y=12  then log3(xy)  =

So what is the general rule?

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