How to do this logarithm proof

[davedave]

This is not a homework question. I just try it for enjoyment.

Let L = log to the base x of (yz) M = log to the base y of (xz) and
N = log to the base z of (xy)

This is how I do it without much luck.

I put all the equations in exponential form

yz = x^L xz = y^M xy = z^N

raise the right-hand sides of the equations to the required power so that x y z will have a product of LMN in the exponent.

(yz)^(MN) = x^(LMN) (xz)^(LN) = y^(LMN) (xy)^(LM) = z^(LMN)


combining the equations gives (yz)^(MN) * (xz)^(LN) * (xy)^(LM) = (xyz)^(LMN)

now, multiply both sides of the equation to (xyz)^-2 and rearrange the terms

x^(L(M+N)-2) * y^(M(L+N)-2) * z^(N(L+M)-2) = (xyz)^(LMN-2)

But, if I could make the left side of the equation above into (xyz)^(L+M+N), then that would complete the proof.

Can someone think of any other way of doing this proof? Thanks.

sorry about my omission.

Here is the question.

Prove that L + M + N = LMN - 2

 

[rock.freak667]

What exactly is the proof you are looking to do?

 

[Ray Vickson]

The equation x^L = y*z gives L = ln(y*z)/ln(x) = [ln(y) + ln(z)]/ln(x), etc.

RGV