1. ## Logs

Calculate the ratio x/y if 2log5(x-3y)=log5(2x)+log5(2y)

2. Originally Posted by Tiger
Calculate the ratio x/y if 2log5(x-3y)=log5(2x)+log5(2y)
$2\log_{5}(x-3y)=\log_{5}(2x)+\log_{5}(2y)$
$\log_{5}(x-3y)^2=\log_{5}(4xy)$
$(x-3y)^2=4xy$
Now, devide both sides by 4x to obtain a formula for y.
and then deivde both sides by 4y to obtain a formula for x.
Hence, You got a formula for x and a formula for y.
Then you are able to find the ratio $\frac{x}{y}$.

3. Originally Posted by Tiger
Calculate the ratio x/y if 2log5(x-3y)=log5(2x)+log5(2y)
First note that $x,y > 0$. Now,

$2 \log_5 (x - 3y) = \log_5 (2x) + \log_5 (2y)$

$\Rightarrow \log_5 (x - 3y)^2 = \log_5 (4xy)$

$\Rightarrow (x - 3y)^2 = 4xy$

$\Rightarrow x^2 - 10xy + 9y^2 = 0$

$\Rightarrow (x - 9y)(x - y) = 0$

So that $x - 9y = 0 \implies \boxed{ \frac xy = 9}$

or

$x - y = 0 \implies \boxed{\frac xy = 1}$

4. Originally Posted by General
$2\log_{5}(x-3y)=\log_{5}(2x)+\log_{5}(2y)$
$\log_{5}(x-3y)^2=\log_{5}(4xy)$
$(x-3y)^2=4xy$
Now, devide both sides by 4x to obtain a formula for y.
and then deivde both sides by 4y to obtain a formula for x.
Hence, You got a formula for x and a formula for y.
Then you are able to find the ratio $\frac{x}{y}$.
This doesn't work. you would get x/y = x/y, which isn't helpful