Question

How do you evaluate `log_81 3`?

Answer 1

`log_81 3 = 1/4`

Explanation:

`log_81 3 =x`

Rewrite as an exponential. Remember, the answer to a log is the exponent. In this case `x` is the exponent, and `81` is the base.

`81^x =3`

Find a common base for both sides, which is `3`.
`81=3^4`, so by substitution...

`(3^4)^x=3`

Use the exponent rule `(x^a)^b=x^(ab)`

`3^(4x)=3^1`

`4x=1`

`x=1/4`

Answer 2

`log_81(3)=1/4`

Explanation:

Since `81` is much larger than `3`, our answer will be a decimal, so let's think of this problem in the opposite sense: `log_3(81)`. `3^4 = 81`, so `log_3(81) = 4`.

Using the law of exponents, we know that if `a^m = n`, then `n^(1/m) = a`. So using this rule, we know that `81^(1/4) = 3`, so `log_81(3)=1/4`