# Logarithms: equations, systems and properties

- Author:
- JLF

## 1. Properties

**1.**

**Logarithm of a product**

**2.**

**Logarithm of a quotient 3.**

**Logarithm of a power **4.**Change of base** 5. Inverse## 2. Equations

A

**logarithmic equation**is an equation that has an unknown factor in the**argument of a logarithm**. In reality, the resolution is reduced to the resolution of equations of the same type as the expressions in the arguments (quadratic equations, cubic equations, irrational equations...).**Example 1**

**We will use the properties of logarithms and that**

*x = 50*.

**Example 2**

## 3. Logrithmic equation systems

**Example 1** We apply the change of variable This way we obtain the following linear equation system we resolve it and we undo the cange of variable:

**Example 2**

****First we apply the property

*logarithm of a product*: We apply the change of variable: and we obtain the following system: that has as solution: Finally, we undo the change of variable: We do not need to check the solutions.

**More examples:**