Let `lambda=log_(5)log_(5)(3)`. If `3^(k+5^(-lambda))=405`, then the value of k is :

Options

A. 3
B. 5
C. 4
D. 6

Answer

To solve the problem step by step, we start with the given information:

Let \( \lambda = \log_5(\log_5(3)) \).

We are given the equation:

\[ 3^{k + 5^{-\lambda}} = 405 \]

### Step 1: Simplify \( 5^{-\lambda} \)

Using the definition of \( \lambda \):

\[ \lambda = \log_5(\log_5(3)) \]

Thus,

\[ 5^{-\lambda} = 5^{-\log_5(\log_5(3))} = \frac{1}{\log_5(3)} \]

### Step 2: Substitute \( 5^{-\lambda} \) into the equation

Now substitute \( 5^{-\lambda} \) into the original equation:

\[ 3^{k + \frac{1}{\log_5(3)}} = 405 \]

### Step 3: Rewrite 405 in terms of powers of 3

We can express 405 as:

\[ 405 = 3^4 \times 5^1 \]

### Step 4: Take logarithm base 3 of both sides

Taking logarithm base 3 of both sides gives:

\[ k + \frac{1}{\log_5(3)} = \log_3(405) \]

### Step 5: Calculate \( \log_3(405) \)

Using the property of logarithms:

\[ \log_3(405) = \log_3(3^4 \times 5^1) = 4 + \log_3(5) \]

### Step 6: Substitute back into the equation

Now we have:

\[ k + \frac{1}{\log_5(3)} = 4 + \log_3(5) \]

### Step 7: Isolate \( k \)

Rearranging gives:

\[ k = 4 + \log_3(5) - \frac{1}{\log_5(3)} \]

### Step 8: Simplify \( \frac{1}{\log_5(3)} \)

Using the change of base formula:

\[ \log_5(3) = \frac{1}{\log_3(5)} \]

Thus,

\[ \frac{1}{\log_5(3)} = \log_3(5) \]

### Step 9: Substitute back into the equation for \( k \)

Substituting this back gives:

\[ k = 4 + \log_3(5) - \log_3(5) = 4 \]

### Final Answer

The value of \( k \) is:

\[ \boxed{4} \]