Another Logarithm Problem

Date: 10/2/95 at 12:32:32
From: David Farnell
Subject: log question

We are having a very difficult time with this problem. Could you 
help us out?

log  69= log  69
   b        10 
         _______
         log  b
            10

We are supposed to prove this for every positive number b but b 
cannot equal 1.

Thank you for your help.

David Farnell, Math/Science Teacher
North River School District

Date: 10/2/95 at 13:8:32
From: Doctor Andrew
Subject: Re: log question

Why don't we try this as generally as we can.  The general rule is
(I'm using "_" to indicate subscripts and <> to indicate not 
equal):

log_b a = (log_c a) / (log_c b) for a > 0, b,c > 0 and b,c <> 1.

If you can prove this, your example is definitely true.

All you really need for this proof is a little knowledge about 
exponents and the definition of a logarithm which is:

x = log_b a if and only if (written iff) b^x = a

I'll get you started.  Let a >= 0 and b,c > 0 and b,c <> 1.  

Now in general if (thing1 iff thing2) and (thing2 iff thing3) then
(thing1 iff thing2).  Keep this in mind when going about the 
proof.  What you want to show in the end (what you want to be at 
the ends of this chain of if and only ifs) is:

x = log_b a  iff  x = log_c a / log_c b.

I think the easiest way to do this problem is to start with the 
righthand side.  What can you say that is true if and only if 
x = log_c a / log_c b?

Good luck!  If you want some help on the next step, please write 
us back.

-Doctor Andrew,  The Geometry Forum