To solve the problem, we need to identify the base used in a common logarithm.
1. Understanding Common Logarithms:
A common logarithm is a logarithm with base 10. It is usually written as $ \log x $ without explicitly showing the base.
2. Definition:
The expression $ \log x $ refers to $ \log_{10} x $, which means "the power to which 10 must be raised to get $x$."
3. Conclusion:
Since the base of the common logarithm is 10, the correct answer is 10.
Final Answer:
The base of the common logarithm is $ \mathbf{10} $.
Consider two distinct positive numbers \( m, n \) with \( m > n \). Let \[ x = n^{\log_n m}, \quad y = m^{\log_m n}. \] The relation between \( x \) and \( y \) is -
If \[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 16, \] then find the value of the given expression.