Converting Logs to Another Base

Date: 04/27/98 at 01:50:52
From: Ashraf Suleman
Subject: Finding the nth log

Hello,

I'm taking a computer programming course in Grade 12 (Pascal), and on 
the side I've decided to write a calculator program for use in DOS. 
The compiler I'm using, however, only performs logarithms to the 
base e. How can I use it to find log to the base 10, or log to the 
base n (if possible)? I don't need the specific programming code, just 
the method I would use is sufficient. I've searched through the site 
here but I haven't found anything pertaining to this (I've yet to take 
algebra, calculus, or finite, so I don't have a grasp of many of the 
terms). Any help would be greatly appreciated. Thanks :)

Date: 04/27/98 at 07:41:37
From: Doctor Jerry
Subject: Re: Finding the nth log

Hi Ashraf,

If you want y = log_a(w), then:

     a^y = w

     ln(a^y) = ln(w)

     y*ln(a) = ln(w)

     y = log_a(w) = ln(w)/ln(a)

-Doctor Jerry,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 04/29/98 at 07:44:14
From: Doctor Bradley
Subject: Re: Finding the nth log

There is a nice way of converting logarithms from any base to another.
This is very useful, as you have found in trying to write your 
program.

Before showing you the rule for conversion, let me show you one way to
obtain it so that you can see how and why it works. Let's suppose you
want to find the log base n of x; call this y:

     y = log  x
            n

What this really means is that x = n^y.  Now suppose I am your 
computer and can only take logarithms using the base e. I could take 
the base e logarithm of both sides of the equation x = n^y; this would 
give me:

     ln x = ln (n^y)

Writing "ln" means "log base e". Next we make use of the property of
logarithms which says that ln (n^y) = y * ln (n), so our equation 
becomes:

     ln x = y ln n

Now we divide both sides by ln n to get:

    (ln x)/(ln n) = y

Remember that y is the original logarithm we wanted to find using 
base n, so all we have to do is take the natural (base e) log of x 
and divide by the natural log of n. This conversion rule works for any 
base, not just e; the general rule to convert from base n to any 
base b looks like this:

     log  x = (log  x)/(log  n)
        n         b        b

Hope this helps. Good luck with your programming!

-Doctor Bradley,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/