Solve for x. 3^x = 31

Select the correct choice and if necessary, fill in the answer box to complete your choice
A. x=--- (type an integer or decimal. do not round until the final answer. Then round to three decimal places as needed. Use a comma to separate answers as needed.
B. solution is not a real number.

To solve the equation \(3^x = 31\), we can take the logarithm of both sides. Using the natural logarithm (ln), we have:

\[
\ln(3^x) = \ln(31)
\]

Using the power rule of logarithms:

\[
x \cdot \ln(3) = \ln(31)
\]

Now, solve for \(x\):

\[
x = \frac{\ln(31)}{\ln(3)}
\]

Calculating this:

\[
x \approx \frac{3.43399}{1.09861} \approx 3.128
\]

So the solution is:

**A. \(x \approx 3.128\)** (rounded to three decimal places).

If you need any further details or have questions, feel free to ask! Here are some related questions to consider:

1. What are the properties of logarithms used in solving exponential equations?
2. How can you check if the solution is correct?
3. What would happen if the base was a different number, such as 2 instead of 3?
4. How does this method apply to solving \(a^x = b\) for other values of \(a\) and \(b\)?
5. Can you explain how to graph the function \(y = 3^x\) and where it intersects with \(y = 31\)? 

**Tip:** When dealing with exponential equations, using logarithms is often the most effective approach for isolating the variable.

Discover how to solve the exponential equation 3^x = 31 using logarithms. This guide provides a step-by-step solution, detailing the necessary formulas and concepts involved, making it suitable for Grades 9-12.

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