PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

Problem 4: Solve for x in the equation

displaymath63


Answer: x = 2

Solution:

As you know by now, we can only take the logarithm of a positive number. Therefore, we will have to restrict the domain (values of x) so that the original equation is valid.
Step 1: The term tex2html_wrap_inline67 is valid when x is greater than zero, and the term tex2html_wrap_inline69 is valid when x is greater than 1. If we restrict the domain to the set of all real numbers that are greater than 1, both terms will be valid.
Step 2: Simplify the left side of the original equation using Logarithmic Rule 2:

displaymath71

Step 3: Convert the logarithmic term to an exponential term with base 4 and exponent tex2html_wrap_inline73 :

displaymath75

Step 4: Simplify the left side of the above equation:

displaymath77

Step 5: Multiply both sides of the above equation by (x - 1):

displaymath79

Step 6: Expand the left side of the above equation:

displaymath81

Step 7: Subtract x from both sides of the above equation and add 2 to both sides of the above equation:

displaymath83

Check: Let's substitute the value x = 2 in the original equation and determine whether the left side of that equation equals the right side of that equation after the substitution

displaymath85

Since the value of the left side of the equation is tex2html_wrap_inline73 after you substitute the value 2 for x, and the right side of the equal is tex2html_wrap_inline73 , you have proved your answer.

If you would like to work on another problem, click on Problem.

[Back to Rules of Logarithms] [Back to Exponential Functions]

[Algebra]
[Algebra] [Trigonometry] [Complex Variables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Nancy Marcus

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour