Can log(x)log(y) be reduced? I'm currently taking Pre-Calc and am learning about logs. I know that log(xy)=log(x)+log(y), but can log(x)log(y) be reduced further?

Jadon Johnson

Jadon Johnson

Answered question

2022-11-14

Can log ( x ) log ( y ) be reduced?
I'm currently taking Pre-Calc and am learning about logs. I know that log ( x y ) = log ( x ) + log ( y ), but can log ( x ) log ( y ) be reduced further?

Answer & Explanation

tiulowyn9v

tiulowyn9v

Beginner2022-11-15Added 7 answers

The answer to your question is, technically, yes:
log ( x ) log ( y ) = log ( x log ( y ) ) = log ( y log ( x ) )
The above follows from the logarithm property that log ( x y ) = y log ( x )
I think, though, it's worth an explanation of why these rules exist, and for that, I'm going to dive into a bit of a long-winded derivation, but bear with me:
The logarithm is defined to be the inverse of exponentiation; that is, log a ( b ) is defined explicitly to be some value x for which a x = b. So when you have statements like
log a ( x ) + log a ( y ) = log a ( x y )
what you're really saying is (considering log as log a ):
x y = a log ( x ) a log ( y ) = a log ( x ) + log ( y ) = a log ( x y )
and since we know that a b = a c happens if and onlf if b = c, then we get that log ( x ) + log ( y ) = log ( x y ). So we derive the multiplication rule like so:
x y = ( a log ( x ) ) y = a y log ( x )
and in some way, this is intuitive: the sum of logs becomes the log of a product (combining addition into multiplication) and the product of logs becomes the log of a power (combining multiplication into exponentiation). Unfortunately it's not quite as pretty, but that's the way the cookie crumbles.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?