How do I express complicated logs as single logarithms?

The process of transforming a complicated log to a single logarithm depends on the question. However, there are some basic rules and methods that are commonly involved.

Here is an example:

Example

Express 2log₃x – log₃(x + 4) as a single logarithm

Step 1:

Use the power log rule. The coefficient of 2log₃x becomes the power.

log₃x² - log3(x+4)

This makes it easier to compare the logs as you have the same coefficient.

Step 2:

Use the log rule that states subtracting two logs in the same base is the same as division. The subtraction outside can be turned into division inside.

log₃x²/(x+4)

You can do this because both logs have the same base (3)

Tips:

• Make sure the logs contain the same base

• Revise the basic log rules so you can easily apply them