To solve the equation \( 4^x + 2 = 256 \), follow these steps:

1. **Isolate the exponential term**:
   \[
   4^x = 256 - 2
   \]
   \[
   4^x = 254
   \]

2. **Rewrite \( 4^x \) as a power of 2**:
   Since \( 4 = 2^2 \), we can write:
   \[
   (2^2)^x = 2^{2x}
   \]
   Thus, the equation becomes:
   \[
   2^{2x} = 254
   \]

3. **Take the logarithm of both sides**:
   \[
   2x = \log_2(254)
   \]

4. **Solve for \( x \)**:
   \[
   x = \frac{\log_2(254)}{2}
   \]

Using a calculator, \( \log_2(254) \) is approximately 7.988 (since \( 2^7 = 128 \) and \( 2^8 = 256 \)). Therefore:
\[
x \approx \frac{7.988}{2} \approx 3.994
\]

So, the solution is approximately \( x \approx 3.994 \).

Would you like further details or have any questions? Here are some related questions you might consider:

1. How do you solve similar exponential equations?
2. What properties of logarithms are useful in solving equations?
3. Can you explain how to convert between different logarithmic bases?
4. What are the applications of exponential equations in real life?
5. How do you determine the number of solutions for an equation like this? 

**Tip**: Always check your solutions by substituting them back into the original equation!

Discover how to solve the exponential equation 4^x + 2 = 256 step-by-step. This solution involves isolating the exponential term, converting to logarithmic form, and using logarithm properties, making it suitable for high school students in Grades 9-12.

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