Logarithmic Problem

Date: Wed, 22 Feb 95 20:29:39 -800
From: Stan Olson

Can you help me solve this?

log 3^X = 6 + (1/2)log 9^x

Thanks

Date: Wed, 22 Feb 1995 22:08:49 -0500 (EST)
From: Dr. Ken
Subject: Re: logarithm equation

Hello there!

Well, to solve this equation, you'll have to remember a couple of 
properties of logarithms.  Namely, 

Log(x^y) = y*Log(x)
Log(xy) = Log(x) + Log(y).

To solve this equation, we can simplify the left side of the equation 
to get Log(3^x) = x*Log(3).  Similarly, on the right side, we get 
(1/2)Log(9^x) = (1/2)Log(3^2x)
                       = Log((3^2x)^1/2)
                       = Log(3^x)
                       = x*Log(3)

So our equation is x*Log(3) = 6 + x*Log(3).  Subtracting x*Log(3) from 
both sides, we can see that this equation has no solutions in the real 
numbers.

Thanks for the question!

-Ken "Dr." Math