Question

How do I find `log_27 9`?

Answer 1

The answer is `2/3`.

Method 1
`log_27(9)` can be interpreted as "`27` to what power is equal to `9`.

Since `27^(2/3) = 3^2 = 9, log_27(9) = 2/3`.

Method 2
If it is difficult to see directly, another technique is to change the base:

`log_27(9)` can be rewritten as `(log_3(9))/log_3(27) = 2/3`

Answer 2

As has already been pointed out, `log_27(9)` is the exponent needed on 27 to get 9. It is the solution to `27^x=9`.

`27^x=9`

Because `27=3^3` and `9=3^2`, we must want

`(3^3)^x=3^2`

But `(3^3)^x=3^(3x)`, so we also want:

`3^(3x)=3^2`.

Now, it is a property of exponents the: if `3` to this power equals `3` to that power, then this power must be equal to that power.

So we need `3x=2`, which means we need #x=2/3.

Check: `27^(2/3)=root(3)27^2=3^2=9`