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In this lesson, we will look into how to solve logarithmic equations using the Properties of Logarithms.

**Properties of Logarithms - Part 1**
**Properties of Logarithms - Part 2 - Solving Logarithmic Equations**

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

More Lessons for Algebra

Math Worksheets

In this lesson, we will look into how to solve logarithmic equations using the Properties of Logarithms.

An equation that contains a logarithm of a variable quantity is called a logarithmic equation. Logarithmic equations can generally be solved using the properties of logarithm and the following property.

For two logarithms of the same base,

Log

log_{a}M =↔_{a}NM = N

** Example: **

Solve the logarithmic equation

log _{2} (*x* – 1) + log _{2} (*x* – 4) = log _{2} (2*x* – 6)

** Solution: **

log _{2} (*x* – 1) + log _{2} (*x* – 4) = log _{2} (2*x* – 6)

log _{2} (*x* – 1)(*x* – 4) = log _{2} (2*x* – 6)

(*x* – 1)(*x* – 4) = (2*x* – 6)

*x*^{2} – 5*x* + 4 = 2*x* – 6

*x*^{2} – 7 *x* + 10 = 0

(*x* – 2)(*x* – 5) = 0

*x* = 2 or 5

We need to check whether each logarithm is defined for these values of *x*.

When *x* = 2,

log _{2} (*x* – 1) = log _{2 }1

log _{2} (*x* – 4) = log _{2 }(– 2) , whuch is undefined

log _{2} (2*x* – 6) = log _{2} _{ }(– 2) , whuch is undefined

So, *x* = 2 is rejected

When *x* = 5,

log _{2} (*x* – 1) = log _{2 }4

log _{2} (*x* – 4) = log _{2 }1

log _{2} (2*x* – 6) = log _{2} _{ }4

So, *x* = 5 is the answer

**Example:**

Solve the logarithmic equation

log _{3} 2 + log _{3} (*x* + 4) = 2log _{3} *x *

**Solution:**

log _{3} 2 + log _{3} (*x* + 4) = 2log _{3} *x
*log

(

*x* = –2 is rejected because log _{3} (–2) is undefined

So the answer is *x* = 4

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