What is the value of x if `log_(3)x+log_(9)x+log_(27)x+log_(81)x=(25)/(4)?`

Options

A. 9
B. 27
C. 81
D. None of these

Answer

To solve the equation \( \log_{3}x + \log_{9}x + \log_{27}x + \log_{81}x = \frac{25}{4} \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of base 3
We know that:
- \( \log_{9}x = \log_{3^2}x = \frac{1}{2} \log_{3}x \)
- \( \log_{27}x = \log_{3^3}x = \frac{1}{3} \log_{3}x \)
- \( \log_{81}x = \log_{3^4}x = \frac{1}{4} \log_{3}x \)

So we can rewrite the equation as:
\[ \log_{3}x + \frac{1}{2} \log_{3}x + \frac{1}{3} \log_{3}x + \frac{1}{4} \log_{3}x = \frac{25}{4} \]

### Step 2: Combine the logarithmic terms
Let \( y = \log_{3}x \). Then we have:
\[ y + \frac{1}{2}y + \frac{1}{3}y + \frac{1}{4}y = \frac{25}{4} \]

### Step 3: Find a common denominator
The common denominator for \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \) is 12. We can rewrite the equation as:
\[ \frac{12}{12}y + \frac{6}{12}y + \frac{4}{12}y + \frac{3}{12}y = \frac{25}{4} \]

### Step 4: Combine the fractions
Now combine the left side:
\[ \left( \frac{12 + 6 + 4 + 3}{12} \right)y = \frac{25}{4} \]
\[ \frac{25}{12}y = \frac{25}{4} \]

### Step 5: Solve for \( y \)
To isolate \( y \), multiply both sides by \( \frac{12}{25} \):
\[ y = \frac{25}{4} \cdot \frac{12}{25} = \frac{12}{4} = 3 \]

### Step 6: Convert back to \( x \)
Since \( y = \log_{3}x \), we have:
\[ \log_{3}x = 3 \]
This implies:
\[ x = 3^{3} = 27 \]

### Final Answer
Thus, the value of \( x \) is \( 27 \).

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