Question

If \[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 16, \] then find the value of the given expression. 
 

Hint:

When dealing with logarithms having fractional bases like $p^{1/2}$, convert them using exponent rules first, then apply change of base formula.

Answer 1

Correct Answer: 4

Step 1: Use change of base formula.
We use the identity:
\[ \log_a b = \frac{\log b}{\log a}. \] Now,
\[ \log_{p^{1/2}} y = \frac{\log y}{\log p^{1/2}}. \] Since, \[ \log p^{1/2} = \frac{1}{2}\log p, \] we get: \[ \log_{p^{1/2}} y = \frac{\log y}{\frac{1}{2}\log p} = \frac{2\log y}{\log p}. \]
Step 2: Simplify second logarithm.
Similarly, \[ \log_{y^{1/2}} p = \frac{\log p}{\log y^{1/2}}. \] And, \[ \log y^{1/2} = \frac{1}{2}\log y. \] Thus, \[ \log_{y^{1/2}} p = \frac{\log p}{\frac{1}{2}\log y} = \frac{2\log p}{\log y}. \]
Step 3: Multiply both expressions.
Now multiply: \[ \left(\frac{2\log y}{\log p}\right) \times \left(\frac{2\log p}{\log y}\right). \] Cancel $\log y$ and $\log p$:
\[ = 4. \]
Step 4: Compare with given value.
Given: \[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 16. \] But we obtained: \[ 4. \] Hence the constant multiplier must be 4 times larger.
Therefore, the required value is:
\[ \boxed{4}. \]