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### Logs in Different Bases

```
Date: 6/11/96 at 21:40:53
From: Anonymous
Subject: Logs in different bases

Suppose a and b are positive numbers for which

log (a) = log (b) = log  (a+2b).  Find b/a.
9        15         25

- Michael
```

```
Date: 6/12/96 at 5:53:9
From: Doctor Pete
Subject: Re: log problem

Here are some hints:

Let n = log (a) = log (b) = log  (a+2b).
9         15        25

Then 9^n = a, 15^n = b, and 25^n = a+2b.

Factoring the bases, we see that

3^(2n) = a, 3^n * 5^n = b, and 5^(2n) = a+2b.

Look at these relations carefully, and you will find that you can
express two of these in terms of the third, thereby eliminating n.
Once you do this, you have a relation in a and b which allows you to
find b/a.

Another thing to keep in mind: note b/a must be positive; this should
eliminate any extraneous solutions you may obtain.

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents

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