[MATH] log(x+y)/logx = log(x+y)/logy[/MATH]

 

Can anybody show a calculation?

I think that's a trick.

Sorry, what do you actually want to do with that equation?

solve for x in terms of y

Well, that looks pretty trivial from the face of it, but there are some technicalities. I don't know whether that's a typo, but you have the same numerator on both sides, so just cancel that out (unless x+y=1 in which case you're screwed) and then anti-log both sides.

I got log y is positive or negative x....

y can't be negative x otherwise the equation wouldn't hold (log(0) not defined).

Touché!

Cheers I got the solution y=x also.

 

I could swear that the question has a typo in it somewhere, that just seems too simple.

Well, that looks pretty trivial from the face of it, but there are some technicalities. I don't know whether that's a typo, but you have the same numerator on both sides, so just cancel that out (unless x+y=1 in which case you're screwed) and then anti-log both sides.

x+y=1

x=y

 

Ya there are two answers.

But what makes you know that x+y=1 is also possible.

Normally, don't us only cancel the log sign to find out x=y instead of minus the left side by the right side and then fing x+y=1?

Look at the solution :

 

[MATH]\frac{\log(x+y)}{\log(x)} = \frac{\log(x+y)}{\log(y)}[/MATH]

[MATH]\log(x+y) \times (\frac{1}{\log(x)} - \frac{1}{\log(y)}) = 0[/MATH]

[MATH]\Rightarrow \log(x+y) = 0 \ldots or \ldots \frac{1}{\log(x)}=\frac{1}{\log(y)}[/MATH]

[MATH]\Rightarrow x+y=1 \ldots or \ldots \log(x)=\log(y)[/MATH]

[MATH]\Rightarrow x+y=1 \ldots or \ldots x=y [/MATH]

yes I know it.

But why do you use this method?

Normally, don't we use the cross-method calculation only?

If I don't say there has a trick or there are two possible answers, how many answers do you desire to get?

When you cancel out a term from both sides, you always assume that this term is non-zero. Hence you must consider the case of this term being zero seperately. In the particular question this term is [MATH]\log(x+y)[/MATH] and [MATH]x+y=1[/MATH] comes out of the case when it is zero.

sigh. I suck

When you cancel out a term from both sides, you always assume that this term is non-zero. Hence you must consider the case of this term being zero seperately. In the particular question this term is [MATH]\log(x+y)[/MATH] and [MATH]x+y=1[/MATH'] comes out of the case when it is zero.

Thanks.

Is there any other cases for other regions of mathematics?

Is there any other cases for other regions of mathematics?

What do you mean ?

such as normal algebra.