Deriving the Change of Base Formula for Logarithms

Date: 04/13/2007 at 22:02:49
From: Joe
Subject: Formula for changing base of a logarithm- why does this work

In my Pre-Calculus class we have been learning about the properties of 
logarithms.  Since calculators have only two bases--base 10 and base 
e, we have learned how to change the base of a logarithm using this 
formula:

           log x
              b
log x =  ----------
   a       log a
              b

I know how to use this formula, but I have no idea why it works.  Can
someone give me a proof explaining why this works?  Thanks a lot.

Date: 04/13/2007 at 22:31:47
From: Doctor Peterson
Subject: Re: Formula for changing base of a logarithm- why does this work

Hi, Joe.

It's not too hard to prove, though the notation can get messy!  I 
often accidentally derive the formula in the course of solving
logarithmic equations.

Let's start by calling the log we're looking for y:

 log_a(x) = y

Now we write that in exponential form:

  x = a^y

(That is, I raised the base a to the exponent on each side of the
original equation.)

Now we want to solve this equation for y, using only base b logs, not
base a logs.  To do this, we take the log of each side:

  log_b(x) = log_b(a^y)

Now we simplify the right side:

  log_b(x) = y log_b(a)

To get y by itself, we just have to divide both sides by log_b(a):

  log_b(x) / log_b(a) = y

Substituting log_a(x) back in for y we have:

  log_a(x) = log_b(x) / log_b(a)

And we're done!

As you can see, the formula is just the natural result of solving
using an available log; that's why I so often get a result looking
like this, and then slap myself on the head and say, "I coulda used
the base-change formula!".

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/