Question 274525: Hi Tutor,
Can you please explain to me why log(base b)x/log(base b)y does not equal log(base b)x - log(base b)y? Thank you
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! I will use the base of 10 to show you because that's what the calculator uses.
The answer applies to all bases.
The rule states that:
log(x/y) = log(x) - log(y)
The rule does not state that:
log(x)/log(y) = log(x) - log(y)
your second statement is clearly invalid if you look at it the following way.
let a = log(x)
let b = log(y)
log(x)/log(y) = log(x) - log(y) becomes:
a/b = a-b which is clearly not valid.
if we use numbers, you will see that the stated rule is valid while the alternate rule you are asking about is not.
let x = 1000
let y = 10
let x/y = 1000/10 = 100
use your calculator to derive:
log(1000) = 3
log(10) = 1
log(1000/10) = log(100) = 2
the rule states that:
log(x/y) = log(x) - log(y)
putting it into numbers, this means that:
log(1000/10) = log(100) = log(1000) - log(10)
since log(100) = 2 and log(1000) = 3 and log(10) = 1, this equation becomes:
2 = 3-1 which is true.
the rule works.
use the alternate rule and you have:
log(x)/log(y) = log(x) - log(y) which becomes:
log(1000) / log(10) = log(1000) - log(10) which becomes:
3/1 = 3-1 which is false.
the alternate rule you asked about does not work.
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