Logarithmic Equations and Systems

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...).

Before starting the exercises, let's remember the logarithmic properties:

Logarithm of a product:

Logarithm of a quotient:

Logarithm of a power:

Change of base:

Usefull property:

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1. Solving a Logarithmic Equation

Example 1:

We use the logarithmic properties and we write 3 as log(1000) to obtain an equality between logarithms:

The logarithms are worth the same when their arguments (what's inside) are the same

We resolve the equation:

Now we have to prove that for these values of x the arguments are not 0 nor negative.

x+1 = 1001/99 > 0 and x-1 = 1001/99-1 >0

Therefore, it is the solution.

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Example 2:

We equal the arguments (what's inside the logarithm) and we resolve the quadratic equation:

So, the arguments coincide when x = 3 and x = 2, therefore, the logarithms will be the same. But we have to prove that for these values of x the arguments are not 0 nor negative.

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2. Solving a Logarithmic Equation System

Example 1:

We apply the change of variable

This way we obtain the following linear equation system

We resolve it and we undo the change of variable:

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Example 2:

We apply the properties logarithm of a quotient and logarithm of a power:

We apply the following change of variable:

and we obtain the system:

that has as solution

Finally, we undo the change of variable:

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More examples:

Logarithmic Equations and Systems

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