MHB Solve Exponent Problem: 3^x=4^y=12^z

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To solve the equation 3^x = 4^y = 12^z, it is essential to express the relationships between the bases. By taking logarithms, one can derive that 3^(xy/(x+y)) equals 12^(zy/(x+y)). Recognizing that both 3 and 4 are factors of 12 helps simplify the problem, leading to the conclusion that z can be expressed as z = (xy)/(x+y). This relationship highlights the interconnectedness of the variables x, y, and z in the context of exponentiation. The discussion emphasizes the importance of using properties of logarithms and factor relationships to solve complex exponent problems.
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Given [Math]3^x=4^y =12^z[/Math] show that $$z=\frac{xy}{x+y}$$

I've take logs on both sides and find myself going in circles, any hints?
 
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$$3^{xy/(x+y)}=12^{zy/(x+y)}$$

Now try using the fact that $3$ and $4$ are factors of $12$ and that $4^y=3^x$.
 

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