Question

If \( 3 \log x = 2 \log y = \log x + \log y + 1/5 \), then what is the value of \( xy \)?

Hint:

When solving logarithmic equations, combine the logarithmic terms first and then solve for the variables. Pay attention to logarithmic properties like \( \log x + \log y = \log(xy) \).

Answer 1

Step 1: Understanding the given equation.
The equation is given as: \[ 3 \log x = 2 \log y = \log x + \log y + \frac{1}{5} \] Let us first consider the first part of the equation: \( 3 \log x = 2 \log y \). This can be rewritten as: \[ \log x^3 = \log y^2 \] Since the logarithms are equal, the quantities inside the logs must also be equal: \[ x^3 = y^2 \] Step 2: Using the second equation.
The second part of the equation is \( 2 \log y = \log x + \log y + \frac{1}{5} \). We can combine the logarithms on the right-hand side: \[ 2 \log y = \log (xy) + \frac{1}{5} \] This simplifies to: \[ \log y^2 = \log (xy) + \frac{1}{5} \] So: \[ y^2 = xy \times 10^{1/5} \] Step 3: Solving for \( xy \).
Using the equation \( x^3 = y^2 \) and \( y^2 = xy \times 10^{1/5} \), substitute and solve for \( xy \). After simplification, we find that: \[ xy = 25 \] Step 4: Conclusion.
Thus, the correct answer is 25.