Calculus question, probably involving the mean value theorem.

[God]

The question states :

If 1<g<h then there's a real k∈]g;h[ such as ln(g)/ln(h) = exponential( (g-h)/(kln(k))
(g, h, k are all reals)

True or false ?

I assumed it's true ( but perhaps it's not, I have to prove it )
I'm 99% sure the mean value theorem is necessary here so I tried to use it randomly trying with ln :
(hypothesis)

k=(g-h)/(ln(g)-ln(h)), g=/=h

I chose ln because it's continuous and differentiable in ]1; +inf] and g and h are superior to one, so it rang a bell
but trying to solve this with this hypothetic k value didn't work, plus when I tested it on the calculator, results seemed to be wrong

Can I have an idea ? I really don't know where to start.
Thanks

 

[daon2]

1/(kln(k)) = (ln(ln(h)) - ln(ln(g)))/(h-g)

Note the quantity on the RHS is positive, so in particular k>1. The function 1/(xln(x)) can be shown to be 1-1 on (1,infinity). So, if there is such a k, it is unique.

One final hint: take the derivative of ln(ln(x)) and use the mean value theorem