Let `x= 2^(log 3)` and `y=3^(log 2)` where base of the logarithm is 10,then which one of the following holds good.

Options

A. `2x lt y`
B. `2y lt x`
C. `3x=2y`
D. `y=x`

Answer

To solve the problem step by step, we will analyze the expressions for \( x \) and \( y \) and compare them.

### Step 1: Define the values of \( x \) and \( y \)

Given:
\[ x = 2^{\log_{10} 3} \]
\[ y = 3^{\log_{10} 2} \]

### Step 2: Take the logarithm of both sides for \( x \)

Taking logarithm (base 10) of both sides for \( x \):
\[ \log_{10} x = \log_{10} (2^{\log_{10} 3}) \]
Using the property of logarithms \( \log_{10} (a^b) = b \cdot \log_{10} a \):
\[ \log_{10} x = \log_{10} 3 \cdot \log_{10} 2 \]
Let's label this as Equation (1).

### Step 3: Take the logarithm of both sides for \( y \)

Now, taking logarithm (base 10) of both sides for \( y \):
\[ \log_{10} y = \log_{10} (3^{\log_{10} 2}) \]
Using the same property of logarithms:
\[ \log_{10} y = \log_{10} 2 \cdot \log_{10} 3 \]
Let's label this as Equation (2).

### Step 4: Compare the two equations

From Equation (1) and Equation (2), we have:
\[ \log_{10} x = \log_{10} y \]

### Step 5: Conclude the relationship between \( x \) and \( y \)

Since the logarithms are equal, we can conclude that:
\[ x = y \]

### Final Conclusion

Thus, the correct option is:
\[ \text{Option 4: } y = x \]

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