**DEFINITION OF NOTATION: **log_{b} x = y is equivalent
to b^{y} = x

In other words, log _{b} x = y means the same thing as b^{y} =
x

**NOTE:** In the rules below it is assumed that b > 0, b ≠ 1,
M > 0, N > 0, and p is any real number.

I. log _{b}1 = 0

II. log _{b}b = 1

III. log _{b}b^{x} = x

IV.
where x > 0 .

V. log _{b}(M ٠ N) = log_{b}M + log_{b}N

VI.

VII.
[**NOTE: **This rule is often used when solving exponential equations.]

VIII. If log _{b} x = log_{b} y, then x = y . This is true
because f (x) = log_{b} x is a 1 - 1 function.

IX.
[**NOTE**: This is known as the Change of Base Formula.]

X. If log _{b} x = y , then b^{y} = x . This comes from the
basic definition at the top of this page.

XI. If b^{x} = b^{y} , then x = y . This is true because f (x)
= log_{b} x is a 1 - 1 function.

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**NOTE:** Rules III and IV are true becausef (x) = log_{b} x and
f (x) = b^{x} are inverses of each other.

Therefore,(g ◦ f )(x) = x and ( f ◦ g)(x) = x .

**NOTE:** Rules VIII and X are frequently used when solving
log equations.

**NOTE: **Rule XI is sometimes used when solving exponental
equations. It is easy to show that Rule XI

is true, because Rule III is true.