Logarithm of a Power

Logarithm of a Power

How to solve the logarithm of a power: formula, examples, and their solutions.

Logarithm of 1

log_b 1 = 0

logb 1 = 0

This is true because
1 = b0.

This formula is used
to simplify logarthims.

Logarithmic Form

Logarithm of the Base

log_b b = 1

logb b = 0

This is true because
b = b1.

This formula is also used
to simplify logarthims.

Formula

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

logb cn = n logb c

To solve the logarithm of a power,
take the exponent n
out from the log.

Example 1: Simplify log2 8

Simplify the given expression. log_2 8

Change 8 to the power of the base:
23.

Take the exponent 3
out from the log.

Then (given) = 3⋅log2 2.

The logarithm of the base is 1.
So log2 2 = 1.

So (given) = 3⋅1.

3⋅1 = 3

So (given) = 3.

Example 2: Simplify log3 (1/81)

Simplify the given expression. log_3 (1/81)

Change 1/81 to the power of the base:
1/34.

1/34 = 3-4

Negative exponent

Take the exponent -4
out from the log.

Then (given) = -4⋅log3 3.

The logarithm of the base is 1.
So log3 3 = 1.

So (given) = -4⋅1.

-4⋅1 = -4

So (given) = -4.

Example 3: log3 2 = a, log3 32 = ?

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

log3 2 = a
The base of the given log is 2.

So change 32 to the power of 2:
25.

Take the exponent 5
out from the log.

Then (given) = 5⋅log3 2.

log3 2 = a

So (given) = 5⋅a.

So (given) = 5a.