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

In other words:

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=5³  so 5*125=5¹*5³  = 5¹⁺³ =5⁴ .  

But this means  that 

log₅(5*125)=log₅(5⁴) = 4 = 1+3=

log₅(5¹)+log₅(5³)=log₅(5)+log₅(125)

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. 

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=2⁴  so 128/16=2⁷/ 2⁴  = 2^7-4 =2³ .  

But this means  that 

log₂(128/16)=log₂(2³) = 3 = 7-4=

log₂(2⁷)-log₂(2⁴)=log₂(128)-log₂(16)

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.

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

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

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

  M ^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 

log_a(M ^p) = log_a(M)+log_a(M)+log_a(M)+.....+log_a(M)=  p log_a(M)

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

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

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.