To solve for \( x \) in each of the given equations, you can follow the general steps below:

1. **Simplify each side of the equation** by combining like terms or simplifying fractions.
2. **Isolate the exponential term** involving \( x \).
3. **Take the logarithm** (if necessary) of both sides to solve for \( x \).

Let's briefly analyze how to approach the first few problems:

### Example for Equation 27:
\[ \frac{1}{5} \cdot 2^x + \frac{1}{3} \cdot 2^{x+1} = \frac{1}{5} \cdot 2^7 + \frac{1}{3} \cdot 2^8 \]

1. Combine terms with \( 2^x \):
\[
\frac{1}{5} \cdot 2^x + \frac{2}{3} \cdot 2^x = LHS
\]

2. Set \( LHS \) equal to \( RHS \):
\[
\left(\frac{1}{5} + \frac{2}{3}\right) 2^x = \frac{1}{5} \cdot 128 + \frac{1}{3} \cdot 256
\]

3. Simplify fractions and isolate \( x \).

Repeat similar steps for the other equations.

If you'd like, I can demonstrate a detailed solution for a specific equation. Let me know which one you'd like to focus on!