Logarithm of a Quotient

How to solve the logarithm of a quotient (loga x/y): formula, 2 examples, and their solutions.

Formula

Formula

loga x/y = loga x - loga y

You can split the log of a quotient like this.

Example 1

Example

Solution

log2 32/√8 = log2 32 - log28

32 = 25

8 = √23

Power

23 = 23/2

Rational Exponent

log2 25 = 5 log2 2

-log2 23/2 = -[3/2] log2 2

Logarithm of a Power

5⋅1 = 5
-[3/2]⋅1 = -3/2

5 = 10/2

10/2 - 3/2 = 7/2

So 7/2 is the answer.

Example 2

Example

Solution

Every term has log6.
So combine these logs into log6.

First write log6 (.

log6 9

The sign is plus.

So write 9.

-log6 15

The sign is minus.

So divide 15.

+log6 10

The sign is plus.

So multiply 10.

Logarithm of a Product

So
log6 9 - log6 15 + log6 10
= log6 ([9/15]⋅10).

Reduce 10 to, 10/5, 2
and reduce 15 to, 15/5, 3.

9/3 = 3

3⋅2 = 6

log6 6 = 1

So 1 is the answer.