Question 274525
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.