PROOFS OF THE LOGARITHM RULES AND CHANGE-OF-BASE FORMULA

Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

 

PROOF OF PRODUCT RULE:

Let and , then and in exponential form.

Then

 

PROOF OF QUOTIENT RULE:

Let and , then and in exponential form.

Then

 

PROOF OF POWER RULE:

Let and , then and in exponential form.

Then

 

PROOF OF THE CHANGE-OF-BASE FORMULA:

Let y = log_bM. Converting to exponential form, we get M = b^y.

We will now apply "log" (base 10) to both sides of the equality M = b^y. This process preserves equality!!! Note, we could have also used "ln" (base e).

Now we will use the Power Rule for logarithms.

Next, we will isolate the y by dividing both sides by log b.

Finally, we change y back to log_bM, we get

which proves the Change-of-Base Formula.