# How to differentiate a equation with three variables

1. Jun 29, 2008

### ritwik06

1. The problem statement, all variables and given/known data
Find the minimum value of $$|log_{b}a+log_{a}b|$$
where 'a' and 'b' are positive numbers.

First tell me, do I need to differentiate it? If no, then how can this thing be done? I dont know how to differentiate a equation with three variables and that too with modulus involved?

2. Jun 29, 2008

### dynamicsolo

Re: Minima

Yes, this problem can be done, but you will need to rewrite it first. Use the 'change of base formula" to convert $$log_{b} a$$ into an expression in terms of a logarithm base a. (This gives an identity between $$log_{a} b$$ and $$log_{b} a$$ that can be useful to know...)

You will now have a single expression, entirely in terms of log-base-a, which will suggest how to find the minimum of the sum.

(BTW, because of the absolute value sign, there will be one answer for the minimum value, but two answers for the relationship between a and b...)

Last edited: Jun 29, 2008