Question

If $\log_x x - \log_10 \sqrt{x} = 2 \log_10 10$, then the possible value of $x$ is given by:

• A. 10
• B. 1
• C. 100
• D. None of these

Hint:

For logarithmic equations, use properties like change of base and logarithmic identities to solve for $x$.

Answer 1

The Correct Option is A

The given equation is: \[ \log_x x - \log_{10} \sqrt{x} = 2 \log_{10} 10 \] Using the logarithmic properties: \[ \log_x x = 1 \quad \text{and} \quad \log_{10} \sqrt{x} = \frac{1}{2} \log_{10} x \] Substitute into the equation: \[ 1 - \frac{1}{2} \log_{10} x = 2 \times 1 \] Simplifying: \[ 1 - \frac{1}{2} \log_{10} x = 2 \] \[ \frac{1}{2} \log_{10} x = -1 \] \[ \log_{10} x = -2 \] Thus, $x = 10^{-2} = 100$. Thus, the correct value of $x$ is 100.