# How do you solve for x and y if log x - log y =2 and log x + log y = 0?

Sep 1, 2016

The point of intersection is $\left(10 , \frac{1}{10}\right)$, assuming the logarithms are in base $10$.

#### Explanation:

Isolate the $\log x$ in equation $1$.

$\log x = 2 + \log y$

Substitute for $\log x$ in equation $2$:

$2 + \log y + \log y = 0$

$2 + 2 \log y = 0$

$2 \left(1 + \log y\right) = 0$

$\log y = - 1$

$y = {10}^{-} 1$

$y = \frac{1}{10}$

$\therefore \log x = 2 + \log \left(\frac{1}{10}\right)$

$\log x - \log \left(\frac{1}{10}\right) = 2$

$\log \left(\frac{x}{\frac{1}{10}}\right) = 2$

$10 x = {10}^{2}$

$10 x = 100$

$x = 10$

The solution point is $\left(10 , \frac{1}{10}\right)$.

Hopefully this helps!