Intuitive understanding of logarithms I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is log(ab)=log(a)+log(b) So in this way, the logarithm is a fundamental relationship between addition and multiplication. Should logarithms in schools be taught this way? Should I think of them primarily in this way? EDIT: This probably related to the fact that the only continuous functions f that satisfy f(x+y)=f(x)f(y) are exponential functions (there are apparently some super-weird non-continuous non-exponential functions that satisfy that multiplicativity but I have no idea what they are).

proximumha

proximumha

Answered question

2022-08-13

Intuitive understanding of logarithms
I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is
log ( a b ) = log ( a ) + log ( b )
So in this way, the logarithm is a fundamental relationship between addition and multiplication. Should logarithms in schools be taught this way? Should I think of them primarily in this way?
EDIT: This probably related to the fact that the only continuous functions f that satisfy f ( x + y ) = f ( x ) f ( y ) are exponential functions (there are apparently some super-weird non-continuous non-exponential functions that satisfy that multiplicativity but I have no idea what they are).

Answer & Explanation

pelvogp

pelvogp

Beginner2022-08-14Added 18 answers

The property you posted is derived from the exponent property
( a n ) ( a m ) = a n + m
therefore we can see that logarithms and exponents are essentially the same, however in a different notation. I think that makes more sense than them being a relationship between multiplication and addition.
If:
10 n = A
Then:
l o g ( A ) = n
And if:
10 m = B
Then:
l o g B = m
If we take
10 c = ( 10 n ) ( 10 m ) = 10 ( m + n )
It follows to say that:
c = n + m
therefore:
l o g ( A B ) = l o g A + l o g B

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