if `log_(16)25 = k log_(2)5 `then k = _____

Answer

To solve the equation \( \log_{16} 25 = k \log_{2} 5 \), we will follow these steps:

### Step 1: Rewrite the logarithm
We start with the equation:
\[ \log_{16} 25 = k \log_{2} 5 \]
We can express \( 16 \) as \( 2^4 \) and \( 25 \) as \( 5^2 \):
\[ \log_{16} 25 = \log_{2^4} 5^2 \]

### Step 2: Apply the change of base formula
Using the change of base formula, we can rewrite the logarithm:
\[ \log_{a^b} c^d = \frac{d}{b} \log_a c \]
So we have:
\[ \log_{2^4} 5^2 = \frac{2}{4} \log_2 5 = \frac{1}{2} \log_2 5 \]

### Step 3: Substitute back into the equation
Now substituting back into the original equation:
\[ \frac{1}{2} \log_2 5 = k \log_2 5 \]

### Step 4: Isolate \( k \)
To isolate \( k \), we can divide both sides by \( \log_2 5 \) (assuming \( \log_2 5 \neq 0 \)):
\[ k = \frac{1}{2} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{2}} \]