Expressing logarithms as ratios of natural logarithms log 2

starbright49ly

starbright49ly

Answered question

2022-05-20

Expressing logarithms as ratios of natural logarithms
log 2 x log 3 x = ln x ln 2 ÷ ln x ln 3
Why can logarithms be written as ratios of natural logarithms?
Can you explain it abstractly, please?
Example of an abstract explanation: the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function.
The teacher doesn't go into abstractions in class, so I would really like to understand it in an abstract sense. Thank you.

Answer & Explanation

iberistazi

iberistazi

Beginner2022-05-21Added 10 answers

Consider y = log b x. Then, by definition, b y = x and so y ln b = ln x. Thus,
log b x = ln x ln b
A more sophisticated argument is the following. Consider a continuous function L : R + R such that L ( x y ) = L ( x ) + L ( y ). Then F ( x ) = L ( e x ) satisfies F ( x + y ) = F ( x ) + F ( y ) and is thus a continuous automorphism of the additive group R . It is easy to see that F must be a scaling:
F ( x ) = a x for some a. Of course, a = F ( 1 ) = L ( e ). When L ( x ) = log b x we have a = log b e = 1 ln b . Since L ( x ) = F ( ln x ), we have, as before:
log b x = ln x ln b

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