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.