Expand log(x^3 y)

Step by step

The question

Question: expand log(x³y).

Expand means: write it as single logs, added and taken away.

The three log rules.

Times inside becomes plus outside.

Divide inside becomes minus outside.

A power inside comes down in front.

Careful!

The rules only work for times, divide and powers.

There is no rule for log of (a plus b).

1 Split the times and divide

The top is x³ and y multiplied: each gets its own log, with plus.

2 Bring the powers down

Each power comes down in front, as times: 3 log x.

So it is 3 log x + log y.

3 Check

Let's check with numbers: x = 3, y = 2.

With a calculator, the question is about 1.732.

Our answer gives about 1.732 too.

It works!

Let's look back.

First: split the times and divide.

Then: bring the powers down.

Last: check.