Properties of Logarithms -- Part I

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.

Let's use this property to understand logarithms of products, quotients, and powers.

For example:  What is log5(5*125)?  

Notice that 5=51 and 125=5so 5*125=51*5 = 51+3 =5.  

But this means  that 

log5(5*125)=log5(54) = 4 = 1+3=


In other words, the log of the product 5*125 equals  the sum of the logs of 5 and 125.


Apply this idea to the following.  Of course, there is nothing special about the base 5. 

If log232=5 and log264=6  then log2(32*64)  =

If log3 x=8 and log3 y=11  then log3(xy)  =

So what is the general rule?

We can  understand the logarithm of a quotient the same way.  For example:  How is log2(128/16)  related to log2(128)  and  log2(16)?

Well, 128=27 and 16=2so 128/16=27/ 2 = 27-4 =2.  

But this means  that 

log2(128/16)=log2(23) = 3 = 7-4=


In other words, the log of the quotient 128/16 equals  the difference of the logs of 128 and 16.   

Apply this idea to the following.

If log2256=8 and log232=5  then log2(256/32) =

If log3 x=7.3 and log3 y=2.1  then log3(x/y) =

Think of the general rule and check by clicking the button.

Finally, let's look at the logarithm of a power using 

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

Here's one way to start understanding this.  If is a positive whole number, then 

  p = M M M M M .... M    (the number of   M's  that we multiply together on the right side is p.)  Since we already know that the log of  a product is the sum  of the logs of the things we are multiplying,  we find that 

loga(M p) = loga(M)+loga(M)+loga(M)+.....+loga(M)=  p loga(M)

since we are adding  loga(M)  a total of p times.   For example, 

log3(9)= 2  so  log3(95) = 5*2=10.      Now try these. 

  log2256=8 so what is the value of  log2(2565) ? Answer   =

If log3 x=7.1,   what is the value of  log3 (x4)?  Answer   =

Now, it turns out that the same rule works even if  p  is not a whole positive number --- in fact,  it works for any value of  p .  So, once again, guess the general rule for logs of powers and click below to check.


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