Solve. Round your answer to the nearest thousandth.
26=3^x
x=

To solve the equation \(26 = 3^x\), we'll use logarithms. Here's the step-by-step process:

1. Take the natural logarithm (ln) of both sides:

\[
\ln(26) = \ln(3^x)
\]

2. Use the logarithmic property \( \ln(a^b) = b\ln(a) \):

\[
\ln(26) = x \ln(3)
\]

3. Solve for \(x\):

\[
x = \frac{\ln(26)}{\ln(3)}
\]

Now, calculate the values of the natural logarithms:

\[
\ln(26) \approx 3.258
\]
\[
\ln(3) \approx 1.099
\]

So:

\[
x \approx \frac{3.258}{1.099} \approx 2.963
\]

### Final Answer:
\[
x \approx 2.963
\]

Would you like further details or explanations?

Here are five related questions:
1. How do we use logarithms to solve exponential equations?
2. What is the difference between natural logarithms and logarithms with base 10?
3. Can this problem be solved using logarithms with a different base?
4. What are some other methods to solve for exponents?
5. How does the accuracy of logarithmic approximations affect the final result?

**Tip:** When solving exponential equations, logarithms are essential tools to "bring down" the exponent and make solving for it possible.

Solve. Round your answer to the nearest thousandth. 26=3^x x=

Learn how to solve the exponential equation 26 = 3^x using logarithms. This step-by-step guide covers key concepts such as the natural logarithm and the logarithmic property. Suitable for high school students in Grades 9-12, with a solution rounded to the nearest thousandth (x ≈ 2.963).

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