Question

Solve the simultaneous equations:

log x to base of y = 2, 5y = x + 8 log y to base of x

Answer 1

\[\log_{y} x = 2, 5y = x + 8\log_{x} y\]

Answer 2

First we should try to get rid of the logs. Using the fact that:\[\log_b a=x \iff b^x =a\]you can use the first equation to show that:\[\log_y x=2\iff y^2=x\]

Answer 3

If y^2=x, then we also get that:\[y=\sqrt{x}\]Therefore:\[\log_x y=\log_x \sqrt{x}=\frac{1}{2}\log_x x=\frac{1}{2}\]So outr system of equations is really:\[x=y^2\]\[5y=x+4\]which you can solve by normal means.

Answer 4

Make sure to test the answers you get back in the original system. Some of the solutions might be extraneous.