Solve the simultaneous equations
2x^2 = 4y and logxy = 3/2
Explanations & Calculations
"\\qquad\\qquad\n\\begin{aligned}\n\\small 2^{x^2}&=\\small 4^y\\\\\n\\small 2^{x^2}&=\\small (2^2)^y= 2^{2y}\\\\\\\\\n\\text{By equaling the indeces we get}\\\\\\\\\n\\small x^2&=\\small 2y\\cdots\\cdots\\cdots\\cdots(1)\\\\\\\\\n\n\\small log_x\\,y&=\\small \\frac{3}{2}\\\\\n\\small y&=\\small x^{\\frac{3}{2}}\\cdots\\cdots\\cdots\\cdots(2)\\\\\\\\\n\n\\small\\text{By (1)=(2),}\\\\\\\\\n\n\\small x^2 &=\\small 2x^{\\frac{3}{2}}\\\\\n\\small x^2-2x^{\\frac{3}{2}}&=\\small 0\\\\\nx^{\\frac{3}{2}}(x^{\\frac{1}{2}}-2)&=\\small 0\\\\\n\\small x^{\\frac{3}{2}} = 0\\qquad &or\\qquad x^{\\frac{1}{2}}-2=0\\\\\n\\small x= 0\\qquad &or\\qquad x =2^2=4 \\\\\\\\\n\n\\small \\text{accordingly the y values by (2),}\\\\\\\\\n\n\\small y = 0\\qquad & or \\qquad y= 8\\\\\\\\\n\n\\small \\therefore (x,y) = (0,0)\\qquad &or\\qquad (4,8)\n\n\\end{aligned}"
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