To solve the equation \(3^x = 8\), we need to find the value of \(x\). 

Since 8 is not a power of 3, we can solve this equation using logarithms. Here are the steps:

1. Take the natural logarithm (or logarithm to any base) of both sides:
   \[
   \ln(3^x) = \ln(8)
   \]

2. Use the power rule of logarithms, which states \(\ln(a^b) = b \ln(a)\):
   \[
   x \ln(3) = \ln(8)
   \]

3. Solve for \(x\):
   \[
   x = \frac{\ln(8)}{\ln(3)}
   \]

Using a calculator to find the natural logarithms:
- \(\ln(8) \approx 2.0794\)
- \(\ln(3) \approx 1.0986\)

Therefore,
\[
x \approx \frac{2.0794}{1.0986} \approx 1.8928
\]

So, the value of \(x\) that satisfies \(3^x = 8\) is approximately \(1.8928\).

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the general method for solving exponential equations?
2. How can we solve \(2^x = 10\)?
3. How do logarithms help in solving exponential equations?
4. What is the relationship between logarithms and exponents?
5. Can we use a different logarithm base to solve \(3^x = 8\)?
6. How would you solve \(3^x = 8\) using common logarithms (base 10)?
7. What are some real-life applications of solving exponential equations?
8. What are the properties of logarithms that are useful in solving equations?

**Tip:** Remember that logarithms are powerful tools for solving equations involving exponential expressions, and understanding their properties can greatly simplify complex calculations.

Learn how to solve the equation 3^x = 8 using logarithms with a detailed step-by-step solution. This problem covers key concepts such as exponential functions and logarithms, specifically the power rule of logarithms. Suitable for high school students in Grades 10-12.

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