Help with simple logarithmic system of equations

Question

It's simple, I just need a little bit of help to get started. $$\log_yx-\log_xy=\frac{8}{3}$$ $$xy=16$$

Answer 1

Guide to get you started:

• write $\log_x y = \frac{1}{\log_y x}$ and then perhaps some quadratic formula might help to solve for $\log_x y$.

Answer 2

Hint:

First just solve $1$. equation to get some more explicit connection between $x$ and $y$.

Change the base of logarithm to some neutral base and make some substitution.

Answer 3

Hint:

With $u:=\dfrac{\log_2 x}{\log_2 y}, v:=\log_2 y$, we have the system

$$\begin{cases}u-\dfrac1u=\dfrac83,\\uv+v=4.\end{cases}$$

The first equation can be reduced to a quadratic one and solved for $u$, then the second gives $v$.

$u=-\dfrac13,v=6$ or $u=3,v=1$.