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 so x/y = 10^a
log y - log z = 15 so y/z = 10^15
log z - log x = -7 so z/x = 10^-7 so z = x * 10^-7
y/(x * 10^-7) = 10^15
y/x * 10^7 = 10^15
y/x = 10^8
x/y = 10^-8
10^-8 = 10^a
a = -8