Solving Logarithmic Equations

How to solve equations containing log terms in same and different bases?

Solving equations containing logarithmic terms requires a good understanding of logarithm properties and careful handling of bases. The approach differs slightly depending on whether all log terms have the same base or different bases.

The following diagrams show examples of solving equations using the power rule for logs. Scroll down the page for more examples and solutions on solving equations using logs.


 

Logarithm Worksheets
Practice your skills with the following worksheets:
Printable & Online Logarithm Worksheets

General Principles for Solving Logarithmic Equations:
1. Understand Logarithm Definition:
y = log_bx ⇔ b^y = x. This is the key to converting between log and exponential forms.
2. Domain Restriction:
The argument of a logarithm must always be positive. That is, if you have log_bA, then A > 0. This is crucial for checking for extraneous solutions.
3. Logarithm Properties:
log_b(MN) = log_bM + log_bN (Product Rule)
log_b(M/N) = log_bM - log_bN (Quotient Rule)
log_b(M^p) = p log_b M (Power Rule)
log_bb = 1
lob_b1 = 0
4. One-to-One Property:
If log_bM = log_bN, then M = N.

Case 1: Equations with Log Terms in the Same Base
This is the more straightforward case. The general strategy is to condense the logarithms on each side (if there are multiple) and then either convert to exponential form or use the one-to-one property.

Case 2: Equations with Log Terms in Different Bases
When you have logarithms with different bases, your primary tool is the Change of Base Formula.

A-Level Maths : Logarithms : Equations 1
In this tutorial I show you how to solve equations where the unknown is a power by using the power rule for logarithms

• Show Step-by-step Solutions

A-Level Maths : Logarithms : Equations 2
In this tutorial I show you how to solve equations containing log terms in the same base.

• Show Step-by-step Solutions

A-Level Maths : Logarithms : Equations 3
In this tutorial I show you how to solve equations containing log terms in different bases.

• Show Step-by-step Solutions

A-Level Maths : Logarithms : Equations 4
In this tutorial I show you how to solve quadratic equations in exponential terms which are solved using logs.

• Show Step-by-step Solutions

Check out our most popular games!

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.