# Exponential equation

## 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:
$$x=\frac{3^xy}{2^y}$$
$$6^y4=36^x$$

Related Precalculus Mathematics Homework Help News on Phys.org
HallsofIvy
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.

This system of equations have no solution?

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

The actual problem was:

"[URL [Broken] 0012.jpg"]Here.[/URL]

But I simplify it to the one above.

Last edited by a moderator:
symbolipoint
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.

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

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

Use Hall of Ivy substitution

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

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

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

But where I will go out of herE?