# Logarithms: equations, systems and properties

## 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

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 ﻿ We have an equality between logarithms, so the arguments (what is inside) has to be the same: ﻿ The solution to the equation is 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: