Lesson Change of Base Formula for logarithms

Algebra ->  Algebra  -> Logarithm Solvers, Trainers and Word Problems -> Lesson Change of Base Formula for logarithms      Log On

Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

   

This Lesson (Change of Base Formula for logarithms) was created by by ikleyn(4) About Me : View Source, Show
About ikleyn:

Change-of-Base Formula for logarithms



The definition of the logarithm is given in the lesson WHAT IS the logarithm in this site.
Now we consider the Change-of-Base Formula for logarithms.
The Change-of-Base Formula for Logarithms is
log%28a%2Cx%29+=+log%28b%2Cx%29%2Flog%28b%2Ca%29

Surely, it is assumed that x, a and b are positive real numbers, a and b are different from 1.

Examples
1) According to the Change-of-Base Formula for Logarithms, log%282%2C16%29+=+log%284%2C16%29%2Flog%284%2C2%29+=+2%2F0.5+=+4.
Verify it by making the direct calculation: log%282%2C16%29+=+4. You get exactly the same number as the Change-of-Base Formula for Logarithms produces!

2) According to the Change-of-Base Formula for Logarithms, log%284%2C16%29+=+log%282%2C16%29%2Flog%282%2C4%29+=+4%2F2+=+2.
Check it by making the direct calculation: log%284%2C16%29+=+2. You get the same result as the Change-of-Base Formula for Logarithms produces.

Proof of the Change-of-Base Formula for Logarithms

Let's denote y=log%28a%2Cx%29. Then x=a%5Ey due to the logarithm definition (see the lesson WHAT IS the logarithm).
Since a and b are positive, there is a real number c such as a+=+b%5Ec.
Due to the logarithm definition, c=log%28b%2Ca%29.
Substituting a+=+b%5Ec to expression x=a%5Ey, you get x=b%5Ecy.
Due to the logarithm definition, this means that log%28b%2Cx%29=cy.
Substituting c=log%28b%2Ca%29 and y=log%28a%2Cx%29 to the last expression, you get log%28b%2Cx%29=log%28b%2Ca%29%2Alog%28a%2Cx%29,
which is exactly the required formula log%28a%2Cx%29+=+log%28b%2Cx%29%2Flog%28b%2Ca%29.
(Note that log%28b%2Ca%29 is not equal to zero, because a is not equal to 1).

The Change-of-Base Formula for Logarithms implies that
log%28a%2Cb%29+=+1%2Flog%28b%2Ca%29.
It is obtained from the Change-of-Base Formula for x=b.
This lesson has been accessed 1038 times.