If `log_(10)x,log_(10)y,log_(10)` z are in AP then x, y, z are in

Options

A. AP
B. GP
C. HP
D. None of these

Answer

To determine the relationship between \(x\), \(y\), and \(z\) when \( \log_{10} x\), \( \log_{10} y\), and \( \log_{10} z\) are in Arithmetic Progression (AP), we can follow these steps:

### Step 1: Understand the condition of AP
If \( \log_{10} x\), \( \log_{10} y\), and \( \log_{10} z\) are in AP, it means that:
\[ 2 \log_{10} y = \log_{10} x + \log_{10} z \]

### Step 2: Use properties of logarithms
Using the properties of logarithms, we can rewrite the equation:
\[ \log_{10} y^2 = \log_{10} (xz) \]
This implies:
\[ y^2 = xz \]

### Step 3: Rearranging the equation
From the equation \(y^2 = xz\), we can express it in a different form:
\[ \frac{y}{x} = \frac{z}{y} \]
This indicates that the ratio of \(y\) to \(x\) is equal to the ratio of \(z\) to \(y\).

### Step 4: Conclusion about the relationship
The above ratio condition indicates that \(x\), \(y\), and \(z\) are in Geometric Progression (GP).

### Final Answer
Thus, if \( \log_{10} x\), \( \log_{10} y\), and \( \log_{10} z\) are in AP, then \(x\), \(y\), and \(z\) are in GP.

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