How to find a rational number from an exponent

When you have a logarithm in the form

,

it is equal to

.

Using the information given, we can rewrite the given equation in the second form to get

.

Now solving for we get the result.

When you have a logarithm in the form

,

it is equal to

.

We can rewrite the given equation as

Solving for , we get

.

To identify which answer is incorrect we need to do each of the conversions.

First lets look at

.

Therefore this conversion is true.

Next lets look at . For this particular one we can recognize that anything raised to a zero power is just one therefore this conversion is true.

From here lets look at

Thus

. Therefore this is an incorrect conversion and thus our answer.

The rule for converting exponents to rational numbers is: .

Even with this, it is easier to work the problem as far as we can with exponents, then switch to rational expression when we run out of room:

At last, we convert, and obtain .

Thus,

.

When you have a logarithm in the form

,

it is equal to

.

We can rewrite the given equation as

Solving for , we get

.