To solve the equation \( 3^{3x} + 2 = 256 \), follow these steps:

### Step 1: Isolate the exponential term
Subtract 2 from both sides:
\[
3^{3x} = 256 - 2
\]
\[
3^{3x} = 254
\]

### Step 2: Solve for \( x \)
Take the logarithm of both sides to solve for \( x \). Using the natural logarithm:
\[
\ln(3^{3x}) = \ln(254)
\]

Apply the logarithm power rule \( \ln(a^b) = b \ln(a) \):
\[
3x \cdot \ln(3) = \ln(254)
\]

Now, solve for \( x \):
\[
x = \frac{\ln(254)}{3 \ln(3)}
\]

### Step 3: Calculate the value of \( x \)
Now compute the values:
\[
x \approx \frac{\ln(254)}{3 \times \ln(3)} \approx \frac{5.5373}{3 \times 1.0986} \approx \frac{5.5373}{3.2958} \approx 1.68
\]

So, \( x \approx 1.68 \).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you solve \( 2^{2x} + 5 = 100 \)?
2. Can you solve \( 4^{x+1} = 128 \) for \( x \)?
3. What is the logarithmic form of the equation \( 5^{2x-1} = 625 \)?
4. How does changing the base in an exponential equation affect the solution?
5. What are the properties of logarithms that are useful in solving exponential equations?

**Tip:** Always isolate the exponential term before applying logarithms for a cleaner solution path.

Learn how to solve the exponential equation 3^(3x) + 2 = 256 with a detailed step-by-step solution. This problem involves exponential equations and logarithms, focusing on isolating the exponential term and applying logarithms to find the value of x. Suitable for students in Grades 10-12.

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