If `log_8 x+log_4 x+log_2 x=11` then the value of x is

Options

A. 2
B. 4
C. 8
D. 64

Answer

To solve the equation \( \log_8 x + \log_4 x + \log_2 x = 11 \), we can follow these steps:

### Step 1: Change the base of the logarithms
We will convert all logarithms to the same base. Let's convert them to base 2:

\[ \log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{\log_2 x}{3} \]

\[ \log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{\log_2 x}{2} \]

\[ \log_2 x = \log_2 x \]

Now substituting these into the original equation gives:

\[ \frac{\log_2 x}{3} + \frac{\log_2 x}{2} + \log_2 x = 11 \]

### Step 2: Find a common denominator
The common denominator for the fractions is 6. We rewrite each term:

\[ \frac{2\log_2 x}{6} + \frac{3\log_2 x}{6} + \frac{6\log_2 x}{6} = 11 \]

### Step 3: Combine the fractions
Now combine the fractions:

\[ \frac{2\log_2 x + 3\log_2 x + 6\log_2 x}{6} = 11 \]

This simplifies to:

\[ \frac{11\log_2 x}{6} = 11 \]

### Step 4: Eliminate the fraction
Multiply both sides by 6 to eliminate the fraction:

\[ 11\log_2 x = 66 \]

### Step 5: Solve for \( \log_2 x \)
Now divide both sides by 11:

\[ \log_2 x = 6 \]

### Step 6: Convert back to exponential form
To find \( x \), we convert from logarithmic form to exponential form:

\[ x = 2^6 \]

### Step 7: Calculate the value of \( x \)
Calculating \( 2^6 \):

\[ x = 64 \]

Thus, the value of \( x \) is \( 64 \).

---