If `f(x)=log_(2)x`, then `f((2)/(x))+f(x)=`

Options

A. `log_(2)((2)/(x))+log_(2)x`
B. 1
C. `log_(2)((2+x^(2))/(x))`
D. `log_(2)((2)/(x))*log_(2)x`

Explanation

To solve the problem, we need to find the value of \( f\left(\frac{2}{x}\right) + f(x) \) given that \( f(x) = \log_2 x \).

### Step 1: Write down the function
Given:
\[ f(x) = \log_2 x \]

### Step 2: Find \( f\left(\frac{2}{x}\right) \)
Now, we need to find \( f\left(\frac{2}{x}\right) \):
\[ f\left(\frac{2}{x}\right) = \log_2\left(\frac{2}{x}\right) \]

### Step 3: Use the logarithmic property
Using the property of logarithms that states \( \log_a\left(\frac{b}{c}\right) = \log_a b - \log_a c \), we can simplify:
\[ f\left(\frac{2}{x}\right) = \log_2 2 - \log_2 x \]

### Step 4: Simplify \( \log_2 2 \)
Since \( \log_2 2 = 1 \):
\[ f\left(\frac{2}{x}\right) = 1 - \log_2 x \]

### Step 5: Add \( f(x) \) to \( f\left(\frac{2}{x}\right) \)
Now, we add \( f(x) \) to \( f\left(\frac{2}{x}\right) \):
\[ f\left(\frac{2}{x}\right) + f(x) = (1 - \log_2 x) + \log_2 x \]

### Step 6: Combine the terms
The \( -\log_2 x \) and \( +\log_2 x \) cancel each other out:
\[ f\left(\frac{2}{x}\right) + f(x) = 1 \]

### Final Answer
Thus, the final result is:
\[ f\left(\frac{2}{x}\right) + f(x) = 1 \]

---