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.