Exponential and Logarithmic Functions - Solving Logarithm Equations

Interpret the displayed dots as an infinite sum with bases \(2,4,8,\ldots\). Determine whether any real \(x\) satisfies the equation.

If \( \log_{2}{x} + \log_{4}{x} + \log_{8}{x} + \dots = \frac{12}{7}\), find \(x\).