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. 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
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.
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:
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.
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: