Understanding Arithmetic Progression and Logarithms

The question asks us to determine the validity of two statements based on a condition involving terms in an Arithmetic Progression (AP).

Condition from Arithmetic Progression

We are given that the terms $(10+\log_{10} x)$, $(10+\log_{10} y)$, and $(10+\log_{10} z)$ are in Arithmetic Progression.

For any three terms $a$, $b$, and $c$ to be in AP, the middle term is the average of the other two, meaning $2b = a + c$. Applying this rule to our terms:

$ 2(10+\log_{10} y) = (10+\log_{10} x) + (10+\log_{10} z) $

Let's simplify this equation:

$ 20 + 2\log_{10} y = 10 + \log_{10} x + 10 + \log_{10} z $

$ 20 + 2\log_{10} y = 20 + \log_{10} x + \log_{10} z $

Subtracting 20 from both sides gives:

$ 2\log_{10} y = \log_{10} x + \log_{10} z $

Using the logarithm property $n \log a = \log a^n$, we can rewrite the left side:

$ \log_{10} (y^2) = \log_{10} x + \log_{10} z $

Using the logarithm property $\log a + \log b = \log (ab)$, we can rewrite the right side:

$ \log_{10} (y^2) = \log_{10} (xz) $

Since the logarithms are equal and the base is the same, their arguments must be equal:

$ y^2 = xz $

Analysis of Statement I

Statement I says: "The GM of $x$ and $z$ is $y^2$."

The Geometric Mean (GM) of two numbers $a$ and $b$ is defined as $\sqrt{ab}$.

So, the GM of $x$ and $z$ is $\sqrt{xz}$.

From our AP calculation, we found that $xz = y^2$.

Therefore, the GM of $x$ and $z$ is $\sqrt{y^2}$. Assuming $y$ is positive (as $\log_{10} y$ is involved), $\sqrt{y^2} = y$.

The statement claims the GM is $y^2$, but we found it to be $y$. Thus, Statement I is incorrect.

Analysis of Statement II

Statement II says: "The AM of $\log_{10} x$ and $\log_{10} z$ is $\log_{10} y$."

The Arithmetic Mean (AM) of two numbers $a$ and $b$ is defined as $\frac{a+b}{2}$.

So, the AM of $\log_{10} x$ and $\log_{10} z$ is:

$ AM = \frac{\log_{10} x + \log_{10} z}{2} $

From our simplification of the AP condition, we derived:

$ 2\log_{10} y = \log_{10} x + \log_{10} z $

Dividing both sides by 2 gives:

$ \frac{2\log_{10} y}{2} = \frac{\log_{10} x + \log_{10} z}{2} $

$ \log_{10} y = \frac{\log_{10} x + \log_{10} z}{2} $

This shows that the AM of $\log_{10} x$ and $\log_{10} z$ is indeed equal to $\log_{10} y$. Thus, Statement II is correct.

Conclusion

Based on the analysis:

• Statement I is incorrect.
• Statement II is correct.

Therefore, only Statement II is correct.