How do I solve these simultaneous equations?
$2 log_x y+2log_yx = 5$
$xy=8$
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?
Best Answer
Say $\log_x y=a$
Therefore $$a+\frac{1}{a}=\frac{5}{2}$$ $$2a^2-5a+2=0$$ $$(2a-1)(a-2)=0$$ $$a=2,\frac{1}{2}$$
Hence we have $$\log_x y=2,\frac{1}{2}$$
Now $$xy=8$$ $$1+\log_x y= \log_x 8$$ Hence $$\log_x 8-1=2$$ $$\log_x 8=3$$ $$\log_8 x=\frac{1}{3}$$ $$x=8^{\frac{1}{3}}=2$$
Similarly we also have $$\log_x 8-1=\frac{1}{2}$$ $$\log_x 8=\frac{3}{2}$$ $$x=4$$
The solutions hence follow.