Logarithm of a Power

How to solve the logarithm of a power (loga xm): formula, 3 examples, and their solutions.

Log of 1 (loga 1)

Formula

a0 = 1

Zero Exponent

The exponent is 0.

So loga 1 = 0.
The logarithm of 1 is 0.

Logarthmic Form

This formula is used
when solving logarithm problems.

Log of Itself (loga a)

Formula

a1 = a

The exponent is 1.

So loga a = 1.
The logarithm of itself is 1.

This formula is also used
when solving logarithm problems.

Formula

Formula

loga xm = m loga x

Take the exponent m
out from the log.

Example 1

Example

Solution

8 = 23

Power

Take the exponent 3
out from the log.

log2 23 = 3 log2 2

log2 2 = 1

3⋅1 = 3

So 3 is the answer.

This means 23 = 8.

Example 2

Example

Solution

1/81 = 1/34

1/34 = 3-4

Negative Exponent

Take the exponent -4
out from the log.

log3 3-4 = -4 log3 3

log3 3 = 1

-4⋅1 = -4

So -4 is the answer.

This means 3-4 = 1/81.

Example 3

Example

Solution

32 = 25

Take the exponent 5
out from the log.

log3 25 = 5 log3 2

log3 2 = a

So 5 log3 2 = 5⋅a.

5⋅a = 5a

So 5a is the answer.