Question

If \( \log_{10}x - \log_{10}y = 2 \log_{10}x \), then a possible value of \( x \) is given by:

• A. 10
• B. 1/100
• C. 1/1000
• D. None of these

Hint:

When working with logarithmic equations, remember the properties of logarithms to simplify the expressions and solve for the variable.

Answer 1

The Correct Option is B

We are given the equation: \[ \log_{10}x - \log_{10}y = 2 \log_{10}x \] Step 1: Simplify the equation: \[ \log_{10}x - \log_{10}y = 2 \log_{10}x \Rightarrow \log_{10}x - 2 \log_{10}x = \log_{10}y \Rightarrow -\log_{10}x = \log_{10}y \] Step 2: Since \( -\log_{10}x = \log_{10}y \), we have: \[ \log_{10} \frac{1}{x} = \log_{10}y \] Step 3: Therefore: \[ \frac{1}{x} = y \] Step 4: Now, substitute \( y = \frac{1}{x} \) into the equation to find possible values for \( x \). Solving, we get the possible value \( x = \frac{1}{100} \). Thus, the answer is: b. \( \frac{1}{100} \).