Solve for $x$ and $y$ : $3x^{\log 3}=4y^{\log 4}$, $4^{\log x}=3^{\log y}$

Question

My try: By using the above information we can form two equations \begin{align} (\log 3)(\log 3x) &= (\log 4)(\log 4y) \text{ and}\\ (\log x)(\log 4) &= (\log y)(\log 3). \end{align}

But they become too difficult to solve.

I need your help. Any hint or another method to solve this.

Answer 1

Your first transformed equation is wrong: $$ 2x^n = 2(x^n)\neq(2x)^n, $$

So the right logarithming gives: $$ (\log 3)(\log x) + \log 3 = (\log 4)(\log y) + \log 4 $$

If $u=\log x$, $v=\log y$, $a=\log 3$, $b=log 4$, then:

$$ au+a = bv+b,\\ bu = av $$

Multiply first one by $ab$ and let $t=bu=av$: $$ a^2(bu)+a^2b = b^2(av)+ab^2,\\ (b^2-a^2)t = -ab(b-a),\\ (a+b)t = -ab,\\ t=-\frac{ab}{a+b} $$

From that we get $u = -\frac{b}{a+b}$, $v=-\frac{a}{a+b}$. After that you can substitute for $x,y$ back.