If `log_(2)x + log_(4)x + log_(64)x =5`, find x:

Options

A. 8
B. 16
C. 7
D. 2

Answer

To solve the equation \( \log_{2}x + \log_{4}x + \log_{64}x = 5 \), we can follow these steps:

### Step 1: Rewrite the logarithms in terms of base 2
We know that:
- \( \log_{4}x = \log_{2}x^{1/2} = \frac{1}{2} \log_{2}x \)
- \( \log_{64}x = \log_{2}x^{1/6} = \frac{1}{6} \log_{2}x \)

So we can rewrite the equation as:
\[ \log_{2}x + \frac{1}{2} \log_{2}x + \frac{1}{6} \log_{2}x = 5 \]

### Step 2: Combine the logarithmic terms
Let \( y = \log_{2}x \). Then, substituting \( y \) into the equation gives us:
\[ y + \frac{1}{2}y + \frac{1}{6}y = 5 \]

### Step 3: Find a common denominator
The common denominator for 1, 2, and 6 is 6. Thus, we can rewrite the equation as:
\[ \frac{6}{6}y + \frac{3}{6}y + \frac{1}{6}y = 5 \]

### Step 4: Combine the fractions
Now, combine the fractions:
\[ \frac{6y + 3y + 1y}{6} = 5 \]
This simplifies to:
\[ \frac{10y}{6} = 5 \]

### Step 5: Solve for \( y \)
To eliminate the fraction, multiply both sides by 6:
\[ 10y = 30 \]
Now divide by 10:
\[ y = 3 \]

### Step 6: Substitute back to find \( x \)
Recall that \( y = \log_{2}x \), so:
\[ \log_{2}x = 3 \]
This means:
\[ x = 2^{3} = 8 \]

### Conclusion
Thus, the value of \( x \) is \( 8 \).

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