`log_(5 sqrt(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 logarithm using the change of base formula
The change of base formula states that:
\[ \log_a b = \frac{\log_c b}{\log_c a} \]
We can use this to rewrite \( \log_{5\sqrt{5}} 5 \):
\[ \log_{5\sqrt{5}} 5 = \frac{\log 5}{\log (5\sqrt{5})} \]

### Step 2: Simplify the denominator
Next, we need to simplify \( \log (5\sqrt{5}) \):
\[ \log (5\sqrt{5}) = \log (5 \cdot 5^{1/2}) = \log (5^{3/2}) = \frac{3}{2} \log 5 \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our expression:
\[ \log_{5\sqrt{5}} 5 = \frac{\log 5}{\frac{3}{2} \log 5} \]

### Step 4: Simplify the fraction
The \( \log 5 \) in the numerator and denominator cancels out:
\[ \log_{5\sqrt{5}} 5 = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

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

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