`if log_(64) P^(2)= 1""2/3 "then" log_(2) p/16`= ______

Options

A. 16
B. 2
C. 32
D. 1

Explanation

To solve the problem step by step, we will start from the given equation and manipulate it to find the required logarithm.

### Step 1: Understand the given equation
We are given:
\[ \log_{64}(P^2) = \frac{5}{3} \]
This means that:
\[ P^2 = 64^{\frac{5}{3}} \]

### Step 2: Rewrite the base
We know that \(64\) can be expressed as \(2^6\). Thus:
\[ 64^{\frac{5}{3}} = (2^6)^{\frac{5}{3}} = 2^{6 \cdot \frac{5}{3}} = 2^{10} \]
So we have:
\[ P^2 = 2^{10} \]

### Step 3: Solve for \(P\)
Taking the square root of both sides gives us:
\[ P = 2^{10/2} = 2^5 \]

### Step 4: Find \(\log_2\left(\frac{P}{16}\right)\)
Now we need to find:
\[ \log_2\left(\frac{P}{16}\right) \]
Substituting \(P\) into the expression:
\[ \log_2\left(\frac{2^5}{16}\right) \]
Since \(16\) can be expressed as \(2^4\), we rewrite the expression:
\[ \log_2\left(\frac{2^5}{2^4}\right) = \log_2(2^{5-4}) = \log_2(2^1) \]

### Step 5: Simplify the logarithm
Using the property of logarithms:
\[ \log_2(2^1) = 1 \]

### Final Answer
Thus, the value of \(\log_2\left(\frac{P}{16}\right)\) is:
\[ \boxed{1} \]

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