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(B) Given that $x, y, z$ are in $H.P.$,the middle term $y$ is the harmonic mean of $x$ and $z$,so $y = \frac{2xz}{x + z}$.We need to evaluate the expr

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$2\log(x - z)$

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(B) Given that $x, y, z$ are in $H.P.$,the middle term $y$ is the harmonic mean of $x$ and $z$,so $y = \frac{2xz}{x + z}$.We need to evaluate the expression $E = \log(x + z) + \log(x - 2y + z)$.Using the property $\log(a) + \log(b) = \log(ab)$,we get $E = \log[(x + z)(x - 2y + z)]$.Substitute $y = \frac{2xz}{x + z}$ into the expression:$E = \log\left[(x + z)\left(x - 2\left(\frac{2xz}{x + z}\right) + z\right)\right]$$E = \log\left[(x + z)\left(x + z - \frac{4xz}{x + z}\right)\right]$$E = \log\left[(x + z)^2 - 4xz\right]$$E = \log(x^2 + 2xz + z^2 - 4xz)$$E = \log(x^2 - 2xz + z^2) = \log(x - z)^2$Using the property $\log(a^n) = n\log(a)$,we get $E = 2\log(x - z)$.

If $x, y, z$ are in $H.P.$,then the value of the expression $\log(x + z) + \log(x - 2y + z)$ is

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