Logarithm of a Power

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

Logarithm of 1

log_b 1 = 0

This is true because
1 = b⁰.

This formula is used
to simplify logarthims.

Logarithmic Form

Logarithm of the Base

log_b b = 0

This is true because
b = b¹.

This formula is also used
to simplify logarthims.

Formula

log_b cⁿ = n log_b c

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

Example 1: Simplify log₂ 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)

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 = ?

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.

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