Logarithms

Concept:Apply the logarithm product rule to combine the sum, then equate arguments when the bases are equal.Explanation:Given: $\log_{10}(xy) + \log_{10}(yz) = \log_{10}(xyz)$.Using $\log a + \log b = \log(ab)$, we combine:$\log_{10}[(xy)(yz)] = \log_{10}(xyz)$That is $\log_{10}(x y^2 z) = \log_{10}(xyz)$.Since the bases are equal, the arguments must be equal:$x y^2 z = x y z$Assuming $x \neq 0$ and $z \neq 0$ (so that the logarithms are defined), divide both sides by $xz$:$y^2 = y$Then $y^2 - y = 0 \implies y(y-1)=0$, so $y=0$ or $y=1$.But $y$ must be positive for $\log_{10}(xy)$ and $\log_{10}(yz)$ to be defined. $y=0$ makes the arguments $0$, which is invalid. Thus $y=1$.Answer:D. 1