Notice that by the logarithmic identity $ \log(x) - \log(y) = \log\frac{x}{y} $, the equations are equivalent to $ \log\frac{x}{y}=a $, $ \log\frac{y}{z}=15 $, and $ \log\frac{z}{x}=-7 $ respectively. Adding all three equations together yields $ \log\frac{x}{y} + \log\frac{y}{z} + \log\frac{z}{x} = a + 15 - 7 $. From the identity $ \log (x) + \log (y) = \log (xy) $, we obtain $ \log\left(\frac{x}{y}\cdot\frac{y}{z}\cdot\frac{z}{x}\right) = a + 8 $. Cancelations result in $ \log(1) = a + 8 $. Since $ \log(1) = 0 $, we find $ a = \boxed{-8} $.