Solve 9^x = 27 Step by Step | Exponential equations and logarithms

Step by step

The question

Question: solve 9 to the power x = 27.

x is up in the power.

9^x = 27

1 Write both as powers of 3

9 is 3², and 27 is 3³.

We write both as powers of 3.

3 × 3 = 9

3 × 3 × 3 = 27

So 9 becomes 3², and 27 becomes 3³.

(3²)^x = 3³

A power of a power: multiply the powers.

(a^m)ⁿ = a^{m × n}

The powers multiply: 2 times x is 2x.

3^2x = 3³

2 The powers are equal

Both sides are 3 to a power.

Same base, and they are equal.

So the powers must be equal too.

same base → same power

Now we have 2x = 3.

2x = 3

3 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

2x2 = 32

2 divided by 2 is 1.

So just x is left.

So x = 3/2!

x = 32

4 Check

Let's check: put 3/2 in place of x.

Check: 9³² = (3²)³² = 3³ = 27

We get 27, the same as the right side.

So x = 3/2 is right!

Let's look back.

First: write both as powers of 3.

Then: the powers are equal.

Then: remove the 2.

Last: check.

Answer