CK-12-Pre-Calculus Concepts

3.5. Change of Base http://www.ck12.org

3.5 Change of Base

Here you will extend your knowledge of log properties to a simple way to change the base of a logarithm.
While it is possible to change bases by always going back to exponential form, it is more efficient to find out how to
change the base of logarithms in general. Since there are only baseeand base 10 logarithms on a calculator, how
would you evaluate an expression like log 3 12?

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/61000

http://www.youtube.com/watch?v=9kKg19s5b78 James Sousa: Logarithms Change of Base Formula

Guidance

The change of base property states:
logba=loglogxxab
You can derive this formula by converting logbato exponential form and then taking the log basexof both
sides. This is shown below.

logba=y

→by=a

→logxby=logxa

→ylogxb=logxa

→y=loglogxa

xb

Therefore, logba=loglogxxab.
Example A
Evaluate log 3 4 by changing the base and using your calculator.
Solution:
log 34 =loglog 101043 =ln 4ln 3≈ 1. 262
Example B
Prove the following log identity.