Lesson Solving logarithmic equations

Solving logarithmic equations

Logarithmic equations contain the unknown variable under the logarithmic function.

The simplest logarithmic equation has the form

,

where is a real positive number not equal to 1.

To solve the equation , you need
1) to solve the equation ;
2) to check the found roots for conditions , .
    Only those roots that satisfy these conditions, are the solutions of the original equations.
    Other roots are not the solutions of the original equations.

There are two major methods for solving logarithmic equations:
1) A method of transforming the original equation to the form and then replacing it by the equation , and
2) A method that introduces a new variable.

Examples below show how to use these methods.

Example 1

Solve the equation .

Solution
First, pass from the given equation to the equation
.
Then, simplify this equation by moving the terms from the right side to the left side with the opposite signs and collecting the common terms. As a result, you get the quadratic equation
.
You can solve this equation by applying the quadratic formula (see the lesson Introduction into Quadratic Equations)
.
The two roots are and .
Now check these two roots for conditions and .
Value does not satisfy these conditions, while value does.

Answer: .

Exercise 1
Solve yourself the equation .

Example 2

Solve the equation .
Solution
Using logarithm properties (sum of logarithms is equal to the logarithm of the product; see for example the lesson Properties of the logarithm),
you can transform the original equation into the form
,
Then consider the equation
.

Simplify this equation step by step. Open brackets in the left side, move the terms from the right side to the left side with the opposite signs and collect the common terms:
,
.

Eventually, you get the quadratic equation, which you can solve using quadratic formula (see the lesson Introduction into Quadratic Equations)
.
Two roots are and .
Now you should check these two roots for conditions , and .
The value does satisfy these conditions, while the value doesn't.

Answer: The solution is .

Exercise 2
Solve yourself the equation .

Example 3

Solve the equation .
Solution
Let us introduce a new variable .
Then you get the quadratic equation
.
It has the roots , .
Now the problem is reduced to the solution the equations and .
The first of these two equations has the root .
The second equation has the root .

Answer: The equation has two solutions: and .

Exercise 3
Solve yourself the equation introducing new variable.