# ACT Math : How to find a rational number from an exponent

## Example Questions

### Example Question #6 : Exponents And Rational Numbers

Which of the following is a value of  that satisfies ?

Explanation:

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.

### Example Question #1 : How To Find A Rational Number From An Exponent

Solve for :

Explanation:

When you have a logarithm in the form

,

it is equal to

.

We can rewrite the given equation as

Solving for , we get

.

### Example Question #8 : Exponents And Rational Numbers

Solve for :

Explanation:

When you have a logarithm in the form

,

it is equal to

.

We can rewrite the given equation as

Solving for , we get

.

### Example Question #4 : Exponents And Rational Numbers

Converting exponents to rational numbers often allows for faster simplification of those numbers.

Which of the following is incorrect? Convert exponents to rational numbers.

Explanation:

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.

### Example Question #2 : How To Find A Rational Number From An Exponent

Sometimes, seeing rational numbers makes it easier to understand an equation.

Convert the following into a rational number or numbers:

Explanation:

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,

.