To solve the equation $3^x = 4$, follow these steps:

1. Take the natural logarithm (ln) of both sides to deal with the exponent: $\ln(3^x) = \ln(4)$

2. Apply the logarithmic rule $\ln(a^b) = b \ln(a)$: $x \ln(3) = \ln(4)$

3. Solve for $x$: $x = \frac{\ln(4)}{\ln(3)}$

4. Evaluate the result using a calculator:

   • $\ln(4) \approx 1.386$
   • $\ln(3) \approx 1.099$

   So: $x = \frac{1.386}{1.099} \approx 1.261$

Final Answer:

$x \approx 1.261$

Would you like a more detailed breakdown or have any further questions?

Here are some related questions to deepen understanding:

1. How would the solution change if the base were different, e.g., $5^x = 4$?
2. Can you solve $3^x = 10$ using a similar method?
3. What is the general formula for solving $a^x = b$ for $x$?
4. How would you solve $3^x = 4$ using common logarithms ($\log$) instead of natural logarithms ($\ln$)?
5. How can exponential equations be solved when $x$ appears in multiple terms?

Tip: Always use logarithmic rules to simplify exponents—it’s a fundamental tool in solving exponential equations.