Do you know this property of the logarithm?

azabak

Playing around with logarithms I found an interesting property that "log b^n(a^n) = log b(a)". Then I tried to find some kind of proof that this is right and not only a coincidence. Ι made a gereral formula for any value of both n's (α and β) so that "log b^β(a^α) = x". Therefore "a^α = b^(β*x)" ; "a = b^(β*x/α)" ; "log b(a) = β*x/α" ; "x = (α/β)*log b(a)". And therefore "log b^β(a^α) = (α/β)*log b(a)".

Stephen Tashi

It's unclear what that notation is supposed to mean.

Can you write out what "log b^n(a^n)" means in words? Or perhaps master the forums LaTex: http://physicsforums.com/showthread.php?t=546968

HallsofIvy

I think you mean "[itex]log_{b^n}(a^n)= log_b(a)[/itex]". That is, that the logarithm, base [itex]b^n[/itex], of [itex]a^n[/itex] is the same as the logarithm, base b, of a. (Of course, a and b must be positive.)

If [itex]y= log_{b^n}(a^n)[/itex] then [itex]a^n= (b^n)^y= b^{ny}= (b^y)^n[/itex]. Can you complete it now?