If `(log_(5) K) (log_(3) 5) (log_(k) x ) = k `, then the value of x if k = 3 is

Options

A. 20
B. 24
C. 27
D. 29

Answer

To solve the equation \((\log_{5} K)(\log_{3} 5)(\log_{K} x) = K\) for \(x\) when \(K = 3\), we will follow these steps:

### Step 1: Substitute \(K = 3\)
We start by substituting \(K\) with \(3\) in the equation:
\[ (\log_{5} 3)(\log_{3} 5)(\log_{3} x) = 3 \]

### Step 2: Simplify \(\log_{5} 3\) and \(\log_{3} 5\)
Using the change of base formula, we can express \(\log_{5} 3\) and \(\log_{3} 5\) in terms of natural logarithms or common logarithms:
\[ \log_{5} 3 = \frac{\log 3}{\log 5} \]
\[ \log_{3} 5 = \frac{\log 5}{\log 3} \]

### Step 3: Substitute back into the equation
Now, we substitute these values back into the equation:
\[ \left(\frac{\log 3}{\log 5}\right)\left(\frac{\log 5}{\log 3}\right)(\log_{3} x) = 3 \]

### Step 4: Simplify the product
Notice that \(\frac{\log 3}{\log 5} \cdot \frac{\log 5}{\log 3} = 1\):
\[ 1 \cdot (\log_{3} x) = 3 \]
Thus, we have:
\[ \log_{3} x = 3 \]

### Step 5: Convert to exponential form
To find \(x\), we convert the logarithmic equation to its exponential form:
\[ x = 3^3 \]

### Step 6: Calculate the value of \(x\)
Calculating \(3^3\):
\[ x = 27 \]

### Final Answer
The value of \(x\) when \(K = 3\) is:
\[ \boxed{27} \]

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