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: Understanding the Question
The question asks to evaluate the expression $\log_{p^{1/2}} y \times \log_{y^{1/2}} p$.
Note: There seems to be a typo in the question. The expression is given to be equal to 16, but we are asked to find its value. We will proceed by evaluating the expression on the left-hand side.
Step 2: Key Formula or Approach
We will use the change of base formula for logarithms and the power rule.
Change of Base Formula: $\log_a b = \frac{\log_c b}{\log_c a}$
Power Rule: $\log (a^n) = n \log a$
A combined useful identity is: $\log_{a^n} b = \frac{1}{n} \log_a b$
Step 3: Detailed Explanation
Let's simplify each term of the expression separately.
Simplifying the first term:
\[ \log_{p^{1/2}} y \] Using the change of base formula (with a common base like 'e' or '10'):
\[ \log_{p^{1/2}} y = \frac{\log y}{\log(p^{1/2})} \] Applying the power rule to the denominator:
\[ \frac{\log y}{\frac{1}{2}\log p} = \frac{2\log y}{\log p} \] Simplifying the second term:
\[ \log_{y^{1/2}} p \] Similarly, using the change of base formula:
\[ \log_{y^{1/2}} p = \frac{\log p}{\log(y^{1/2})} \] Applying the power rule to the denominator:
\[ \frac{\log p}{\frac{1}{2}\log y} = \frac{2\log p}{\log y} \] Multiplying the simplified terms:
Now we multiply the results of the two simplified terms:
\[ \left(\frac{2\log y}{\log p}\right) \times \left(\frac{2\log p}{\log y}\right) \] We can cancel out the $\log y$ terms and the $\log p$ terms:
\[ = 2 \times 2 = 4 \] Step 4: Final Answer
The value of the expression is 4.
\[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 4 \]