# Solve simultaneous logarithmic equations with different bases? How do I solve these simultaneous equations? 2log_x y+2 log_y x=5 xy=8

Solve simultaneous logarithmic equations with different bases?
How do I solve these simultaneous equations?
$2lo{g}_{x}y+2lo{g}_{y}x=5$
$xy=8$
I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've tried to do
$x=8/y$
and substitute to the first equation, but I still can't seem to solve this. How do I go about in solving these types of equations?
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Samantha Braun
Say ${\mathrm{log}}_{x}y=a$
Therefore
$a+\frac{1}{a}=\frac{5}{2}$
$2{a}^{2}-5a+2=0$
$\left(2a-1\right)\left(a-2\right)=0$
$a=2,\frac{1}{2}$
Hence we have
${\mathrm{log}}_{x}y=2,\frac{1}{2}$
Now
$xy=8$
$1+{\mathrm{log}}_{x}y={\mathrm{log}}_{x}8$
Hence
${\mathrm{log}}_{x}8-1=2$
${\mathrm{log}}_{x}8=3$
${\mathrm{log}}_{8}x=\frac{1}{3}$
$x={8}^{\frac{1}{3}}=2$
Similarly we also have
${\mathrm{log}}_{x}8-1=\frac{1}{2}$
${\mathrm{log}}_{x}8=\frac{3}{2}$
$x=4$
The solutions hence follow.