To solve this problem, we need to simplify the given logarithmic equation:
1. Given Equation:
\( \log_x b - \log_x a = \log_x c - \log_x b \)
2. Applying Logarithmic Properties:
Using the property \( \log_x p - \log_x q = \log_x \left(\frac{p}{q}\right) \), we can rewrite the equation as:
\( \log_x \left(\frac{b}{a}\right) = \log_x \left(\frac{c}{b}\right) \)
3. Equating the Arguments:
Since the logarithms are equal, their arguments must be equal as well. Therefore:
\( \frac{b}{a} = \frac{c}{b} \)
4. Solving for \( ac \):
Cross multiplying the equation gives:
\( b^2 = ac \)
Final Answer:
The correct option is (B) \( b^2 \).
The product of all solutions of the equation \(e^{5(\log_e x)^2 + 3 = x^8, x > 0}\) , is :