Find x if `log_(1//sqrt(2)) (1//sqrt(8)) = log_(2)(4^(x) +1). Log(4^(x+1) +4)`,

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To solve the equation \( \log_{\frac{1}{\sqrt{2}}} \left(\frac{1}{\sqrt{8}}\right) = \log_{2}(4^{x} + 1) \cdot \log(4^{x+1} + 4) \), we will follow these steps:

### Step 1: Simplify the left-hand side
First, we simplify the left-hand side of the equation:

\[ \log_{\frac{1}{\sqrt{2}}} \left(\frac{1}{\sqrt{8}}\right) \]

Using the change of base formula, we can rewrite this as:

\[ \frac{\log\left(\frac{1}{\sqrt{8}}\right)}{\log\left(\frac{1}{\sqrt{2}}\right)} \]