+489 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?
+9466 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
+128408 log x - log y = a
log y - log z = 15
log z - log x = -7
Add the first two equations and we get that
log x - log z = a + 15
Multiplying the third equation by -1 on both sides we have that
log x - log z = 7
This implies that
a + 15 = 7 subtract 15 from both sides
a = - 8
