Logarithms of Powers

Logarithms of Powers

How to solve logarithms of powers problems: formula, proof, examples, and their solutions.

Basic Properties

log_b 1 = 0. log_b b = 1

1 = b0.
So logb 1 = 0.

b = b1.
So logb b = 1.

Use these two properties to simplify logarithms.

Logarithmic Form

Formula

log_b (c^n) = n*(log_b c)

logb cn = n logb c

Take the exponent n out from cn.

Proof

Logarithms of Powers: Proof of the formula

Set c = b(brown).

Log both sides. (base: b)
logb c = (brown)

Logarithmic form

See c = b(brown) again.

Do the n-th power on both sides.
cn = bn⋅(brown)

Log both sides. (base: b)
logb cn = n⋅(brown)

logb c = (brown)
So logb cn = n⋅logb c.

Example 1

Simplify the given expression. log_2 8

Change 8 to the power of the base: 23.

Take the exponent 3 out.

log2 23 = 3⋅log2 2

log2 2 = 1

Example 2

Simplify the given expression. log_3 (1/81)

Change 1/81 to the power of the base: 3-4.

Negative exponent

Take the exponent -4 out.

log3 3-4 = (-4)⋅log3 3

log3 3 = 1

Example 3

If log_3 2 = a, find the value of the given expression. log_3 32

Change 32 to the power of the base: 25.

Take the exponent 5 out.

log3 25 = 5⋅log3 2

log3 2 = a
(given condition)