# Exponential equation

Physicsissuef

## Homework Statement

Solve this system of equations:

$$3^xy=2^yx$$
$$12^xx=3^y4$$

## The Attempt at a Solution

I was solving and came up, till here:
$$x=\frac{3^xy}{2^y}$$
$$6^y4=36^x$$

Homework Helper
Actually I'm surprised you were able to get that far! Most equations that involve variables both "inside" and "outside" transcendental functions cannot be solved in terms of elementary functions.

Once you are at
$$6^y4= 36^x= (6^2)^x= 6^{2x}$$
You can take the logarithm of both sides:
$$y ln(6)+ ln 4= 2x ln 5$$
Where you would go from there, I have no idea.

Physicsissuef
This system of equations have no solution?

Homework Helper
I didn't say that. It said it might not be possible to solve it using elementary functions.

Physicsissuef
The actual problem was:

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But I simplify it to the one above.

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Homework Helper
Gold Member
Try using change of base of the logarithm functions first, and then try. Put then into either base 2 or base 3. I have not tried this in your exercise but believe it's worth trying.

Physicsissuef
I tried on several ways and it didn't worked.

btw- on the first post should be:
$$6^y4=36^xy$$

tmclary
Use Hall of Ivy substitution

in the above equation: (6^2x)y=(6^y)4.

Physicsissuef
Maybe
$$log_66^y4=log_66^2xy$$

$$y+log_64=2x+log_6y$$

But where I will go out of herE?