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### 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
```
Associated Topics:
High School Basic Algebra
High School Logs
Middle School Algebra
Middle School Logarithms

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