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The equation given is:

\[
500(1.26)^n = 26,425
\]

We need to solve for \(n\) by taking the logarithm of both sides. Here's the step-by-step solution:

1. **Divide both sides by 500** to isolate the exponential term:

\[
(1.26)^n = \frac{26,425}{500}
\]

\[
(1.26)^n = 52.85
\]

2. **Take the natural logarithm (or log base 10) of both sides**:

\[
\log((1.26)^n) = \log(52.85)
\]

Using the logarithmic property \(\log(a^b) = b \cdot \log(a)\), this becomes:

\[
n \cdot \log(1.26) = \log(52.85)
\]

3. **Solve for \(n\)**:

\[
n = \frac{\log(52.85)}{\log(1.26)}
\]

Using a calculator:

\[
\log(52.85) \approx 1.723
\]
\[
\log(1.26) \approx 0.1004
\]

Now, divide:

\[
n = \frac{1.723}{0.1004} \approx 17.16
\]

4. **Round to three decimal places**:

\[
n \approx 17.160
\]

So, the value of \(n\) is approximately **17.160**.

Would you like more details, or do you have any questions?

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Here are five related questions:
1. What is the difference between natural logarithms and logarithms base 10?
2. How do you apply logarithmic properties to simplify equations?
3. How would the solution change if the base were different from 1.26?
4. What are some common uses of logarithms in real-world problems?
5. Can you solve exponential equations without using logarithms? If so, how?

**Tip**: Remember that logarithms are powerful for solving equations involving exponentials because they allow you to "bring down" exponents, making it easier to isolate variables.

500(1.26)^n = 26,425. Solve by taking the logarithm of each side. Round answer to 3 decimal places.

Learn how to solve the exponential equation 500(1.26)^n = 26,425 by applying logarithmic properties. Detailed step-by-step solution provided, suitable for high school students (Grades 10-12).

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