IF `lamda^(log_(5)3`=81 ,find the value of `lamda`.

Answer

To solve the equation \( \lambda^{\log_5 3} = 81 \), we will follow a series of steps to isolate \( \lambda \).

### Step-by-Step Solution:

1. **Write the given equation:**
\[ \lambda^{\log_5 3} = 81 \]

2. **Take the natural logarithm (logarithm to the base \( e \)) on both sides:**
\[ \log(\lambda^{\log_5 3}) = \log(81) \]

3. **Use the logarithmic identity \( \log(a^b) = b \cdot \log(a) \):**
\[ \log_5 3 \cdot \log(\lambda) = \log(81) \]

4. **Express \( 81 \) as a power of \( 3 \):**
\[ 81 = 3^4 \quad \Rightarrow \quad \log(81) = \log(3^4) = 4 \cdot \log(3) \]
So, we can rewrite the equation as:
\[ \log_5 3 \cdot \log(\lambda) = 4 \cdot \log(3) \]

5. **Use the change of base formula for logarithms:**
\[ \log_5 3 = \frac{\log(3)}{\log(5)} \]
Substitute this into the equation:
\[ \frac{\log(3)}{\log(5)} \cdot \log(\lambda) = 4 \cdot \log(3) \]

6. **Cancel \( \log(3) \) from both sides (assuming \( \log(3) \neq 0 \)):**
\[ \frac{\log(\lambda)}{\log(5)} = 4 \]

7. **Multiply both sides by \( \log(5) \):**
\[ \log(\lambda) = 4 \cdot \log(5) \]

8. **Use the property of logarithms to rewrite the equation:**
\[ \log(\lambda) = \log(5^4) \]

9. **Since the logarithms are equal, we can equate the arguments:**
\[ \lambda = 5^4 \]

10. **Calculate \( 5^4 \):**
\[ 5^4 = 625 \]

### Final Answer:
\[ \lambda = 625 \]