`log_(5sqrt(5))5` is equal to

Options

A. `2//3`
B. `1//3`
C. `1//2`
D. 2

Answer

To solve the expression \( \log_{5\sqrt{5}} 5 \), we can follow these steps:

### Step 1: Rewrite the Base
The base \( 5\sqrt{5} \) can be rewritten in terms of powers of 5. We know that:
\[ \sqrt{5} = 5^{1/2} \]
Thus,
\[ 5\sqrt{5} = 5 \cdot 5^{1/2} = 5^{1 + 1/2} = 5^{3/2} \]

### Step 2: Apply the Change of Base Formula
Using the change of base formula for logarithms, we have:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
We can use base 5 for our calculation:
\[ \log_{5\sqrt{5}} 5 = \frac{\log_{5} 5}{\log_{5} (5^{3/2})} \]

### Step 3: Simplify the Logarithms
We know that \( \log_{5} 5 = 1 \) (since any log of a number to its own base is 1). Now, we simplify the denominator:
\[ \log_{5} (5^{3/2}) = \frac{3}{2} \log_{5} 5 = \frac{3}{2} \cdot 1 = \frac{3}{2} \]

### Step 4: Combine the Results
Now, substituting back into our expression, we have:
\[ \log_{5\sqrt{5}} 5 = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

### Final Answer
Thus, the value of \( \log_{5\sqrt{5}} 5 \) is:
\[ \frac{2}{3} \]

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