If $x$, $y$, and $z$ are positive real numbers satisfying: \begin{align*} \log x - \log y &= a, \\ \log y - \log z &= 15, \text{ and} \\ \log z - \log x &= -7, \\ \end{align*}where $a$ is a real number, what is $a$?
\( log x - log y = a, log y - log z = 15, log z - log x = -7\)
log z - log x = -7
log y - log z = 15 add these and we get
logy - log x = 8 multiply through by -1
log x - log y = -8 = "a"
