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

. Converting to exponential form, we gety = log_{b}MM = 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 Rulefor logarithms.Next, we will isolate the

by dividing both sides byy.log bFinally, we change

back toy, we getlog_{b}Mwhich proves the Change-of-Base Formula.