If `log_(16) x + log_(x) x + log_(2) x = 14 ` , then x =

Options

A. 16
B. 32
C. 64
D. 256

Answer

To solve the equation \( \log_{16} x + \log_{x} x + \log_{2} x = 14 \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of a common base
We can convert all logarithms to base 2 using the change of base formula:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
Thus, we have:
\[ \log_{16} x = \frac{\log_{2} x}{\log_{2} 16}, \quad \log_{x} x = 1, \quad \log_{2} x = \log_{2} x \]
Since \( \log_{2} 16 = 4 \) (because \( 16 = 2^4 \)), we can rewrite \( \log_{16} x \):
\[ \log_{16} x = \frac{\log_{2} x}{4} \]

### Step 2: Substitute back into the equation
Now substituting these into the original equation:
\[ \frac{\log_{2} x}{4} + 1 + \log_{2} x = 14 \]

### Step 3: Combine like terms
Let's combine the terms involving \( \log_{2} x \):
\[ \frac{\log_{2} x}{4} + \log_{2} x = \frac{\log_{2} x}{4} + \frac{4 \log_{2} x}{4} = \frac{5 \log_{2} x}{4} \]
So the equation becomes:
\[ \frac{5 \log_{2} x}{4} + 1 = 14 \]

### Step 4: Isolate the logarithm
Subtract 1 from both sides:
\[ \frac{5 \log_{2} x}{4} = 13 \]

### Step 5: Multiply by 4 to eliminate the fraction
Multiply both sides by 4:
\[ 5 \log_{2} x = 52 \]

### Step 6: Divide by 5
Now divide by 5:
\[ \log_{2} x = \frac{52}{5} \]

### Step 7: Convert back to exponential form
To find \( x \), we convert the logarithmic equation back to its exponential form:
\[ x = 2^{\frac{52}{5}} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = 2^{\frac{52}{5}} \]

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