Is there a proof that n x </msup> m x </msup> = (

Adan Norton

Adan Norton

Answered question

2022-06-01

Is there a proof that n x m x = ( n x ) ( log ( m n ) / log ( n ) ) ?
This isn't a homework question, just something I'm curious about, but you can treat it that way if you like.
So the other day I was playing with my calculator and I noticed that
2 x 10 x = ( 2 x ) ( log ( 20 ) / log ( 2 ) )
I tried it out with some other numbers and came to the conclusion that
n x m x = ( n x ) ( log ( n m ) / log ( n ) )
So I wanted to see if there is a way to prove that.
I already know that m = n ( log ( m ) / log ( n ) ) and I figured that there must be a relation. So from that I can see that m n = n ( log ( n m ) / log ( n ) ) . However I don't understand why that would mean that ( n m ) x = ( n x ) log ( n m ) / log ( n ) .
Is what I say actually true? How do the powers fit into the proof?

Answer & Explanation

Kolten Bowen

Kolten Bowen

Beginner2022-06-02Added 3 answers

It's true, and you're almost done. Probably recalling
a x y = ( a x ) y = ( a y ) x
is all you need.
Marquis Cooper

Marquis Cooper

Beginner2022-06-03Added 1 answers

n x m x = ( n x ) log m n / log n
will be true
x log n + x log m = x ( log m + log n ) log n log n
which is true
using log ( a x ) = x log a and log a b = log a + log b

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