To solve the equation $3^x = 32$ using logarithms, follow these steps:

1. Start with the equation: $3^x = 32$

2. Take the natural logarithm ($\ln$) of both sides: $\ln(3^x) = \ln(32)$

3. Use the logarithmic power rule ($\ln(a^b) = b \ln(a)$): $x \ln(3) = \ln(32)$

4. Solve for $x$: $x = \frac{\ln(32)}{\ln(3)}$

5. Compute the logarithms (if numerical evaluation is required): $x \approx \frac{\ln(32)}{\ln(3)} \approx \frac{3.4657}{1.0986} \approx 3.15$

Thus, the solution is: $x = \frac{\ln(32)}{\ln(3)} \quad \text{or approximately } x \approx 3.15$

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Would you like more details or have any questions about this process?

Related Questions:

1. How can you solve $2^x = 10$ using logarithms?
2. What is the difference between natural logarithms ($\ln$) and common logarithms ($\log$)?
3. How can you verify the solution for $3^x = 32$ numerically?
4. What is the base change formula in logarithms and how is it used?
5. How do logarithmic functions differ from exponential functions?

Tip:

Always use the logarithmic power rule to simplify expressions involving exponents before solving for variables.