# Exponential Equations

• Introduction

• Properties of Powers

• 25 Resolved Exponential Equations step by step

## Introduction

An exponential equation is one that has exponential expressions, in other words, powers that have in their exponent expressions with the unknown factor x.

In this section, 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.

For example:

$$3^{2x} = 3^6$$

Obviously, the value that x has to take for the equality to be true is 3.

In order to achieve this type of expressions we have to factorize, express all the numbers in the form of powers, apply the properties of powers and write roots as powers.

Sometimes we will need to make a change of variable to transform the equation in a quadratic one.

We can also resolve using logarithms, but we will leave this type of procedures for more difficult equations with different bases in the exponential expressions, making it impossible to use the previous method of equalizing.

For example,

$$3^{x+3} = 5^x$$

which has a real solution, using logarithms of,

$$x = \frac{3 ln(3)}{ln\left(\frac{5}{3}\right)}$$

## Before we start...let's remember the properties of powers

 Product Power Quotient Negative exponent Inverse Inverse of inverse

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