If `log_(k)xlog_(5)k=3`, then find the value of x.

Options

A. A)`k^(5)`
B. B)`5k^(3)`
C. C)243
D. D)125

Answer

To solve the equation \( \log_k x \cdot \log_5 k = 3 \), we can follow these steps:

### Step 1: Use the Change of Base Formula
We know from logarithmic properties that:
\[ \log_a b = \frac{1}{\log_b a} \]
Using this property, we can rewrite \( \log_5 k \):
\[ \log_5 k = \frac{1}{\log_k 5} \]

### Step 2: Substitute into the Original Equation
Substituting this into the original equation gives us:
\[ \log_k x \cdot \frac{1}{\log_k 5} = 3 \]

### Step 3: Simplify the Equation
Multiplying both sides by \( \log_k 5 \) to eliminate the fraction:
\[ \log_k x = 3 \cdot \log_k 5 \]

### Step 4: Use the Power Rule of Logarithms
Using the property of logarithms that states \( a \cdot \log_b c = \log_b(c^a) \):
\[ \log_k x = \log_k(5^3) \]

### Step 5: Set the Arguments Equal
Since the logarithms are equal, we can set the arguments equal to each other:
\[ x = 5^3 \]

### Step 6: Calculate the Value of \( x \)
Calculating \( 5^3 \):
\[ x = 125 \]

Thus, the value of \( x \) is \( 125 \).

### Final Answer
The value of \( x \) is \( 125 \).

---