# Exponential Equations

- Author:
- JLF

An

**exponential equation**is one that has exponential expressions, in other words, powers that have in their exponent expressions with the unknown factor*x*. For example, In this paper, we will resolve the exponential equations without using logarithms. This method of resolution consists in reaching an equality of the exponentials with the same base in order to equal the exponents.## 1. Remember...

Before we start, let's remember the properties of powers:

**Product
Quocient
Inverse
Power
Negative exponent
Inverse of inverse **

## 2. Solved equations

## Equation 1

Taking into account that , we can rewrite the equation as
Therefore, the solution is .

## Equation 2

Taking into account that , we can rewrite the equation as
Then we have the linear equation . Therefor, the solution is .

## Equation 3

Taking into account that
We can rewrite the equation as
We have the common base , but because one of them is squared, we write
Substituting, the equation finishes like
In other words, a quadratic equation:
We multiply the full equation by 9:
We solve it:
Therefore,
So, we obtain
The second option is not possible because it is negative. Therefore,
From where we obtain the only one solution .

## More examples:

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