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
Answered question
2022-11-14
Can be reduced? I'm currently taking Pre-Calc and am learning about logs. I know that , but can be reduced further?
Answer & Explanation
tiulowyn9v
Beginner2022-11-15Added 7 answers
The answer to your question is, technically, yes:
The above follows from the logarithm property that 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, is defined explicitly to be some value for which . So when you have statements like
what you're really saying is (considering as ):
and since we know that happens if and onlf if , then we get that . So we derive the multiplication rule like so:
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.