solve\:for\:n:\:800\left(1.08\right)^n\:<\:760\left(1.02\right)^{\left(4n\right)}

To solve for \( n \) in the inequality

\[
800(1.08)^n < 760(1.02)^{4n},
\]

we can take the following steps:

1. **Divide both sides by 800** to simplify:
   \[
   (1.08)^n < \frac{760}{800} \cdot (1.02)^{4n}.
   \]
   
   Simplify \( \frac{760}{800} \):
   \[
   \frac{760}{800} = 0.95,
   \]
   so we get
   \[
   (1.08)^n < 0.95 \cdot (1.02)^{4n}.
   \]

2. **Rewrite the inequality** as:
   \[
   \frac{(1.08)^n}{(1.02)^{4n}} < 0.95.
   \]

3. **Simplify the left side** by combining the exponents:
   \[
   \left(\frac{1.08}{(1.02)^4}\right)^n < 0.95.
   \]

4. **Calculate \( \frac{1.08}{(1.02)^4} \):**

   Evaluating \( (1.02)^4 \) gives approximately \( 1.08243216 \), so:
   \[
   \frac{1.08}{1.08243216} \approx 0.9975.
   \]

   Thus, we have:
   \[
   (0.9975)^n < 0.95.
   \]

5. **Take the natural logarithm of both sides** to solve for \( n \):
   \[
   n \ln(0.9975) < \ln(0.95).
   \]

6. **Solve for \( n \)** by dividing both sides by \( \ln(0.9975) \), noting that \( \ln(0.9975) \) is negative, which reverses the inequality:
   \[
   n > \frac{\ln(0.95)}{\ln(0.9975)}.
   \]

7. **Calculate the values**:
   - \( \ln(0.95) \approx -0.0513 \).
   - \( \ln(0.9975) \approx -0.0025 \).

   So,
   \[
   n > \frac{-0.0513}{-0.0025} \approx 20.52.
   \]

Since \( n \) must be an integer, the smallest integer satisfying this inequality is

\[
n = 21.
\]

### Final Answer:
\[
n > 20.52 \Rightarrow n = 21.
\]

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. How would the solution change if the growth rates (1.08 and 1.02) were altered?
2. What happens if we change the constant terms (800 and 760) instead?
3. Could this inequality be solved by graphing the functions on each side?
4. How does changing the exponent of 1.02 from \(4n\) to another multiple of \(n\) affect the result?
5. How does taking the logarithm help simplify exponential inequalities?

#### Tip:
For exponential inequalities, taking logarithms helps simplify and linearize the exponents for easier solving.

solve for n: 800(1.08)^n < 760(1.02)^(4n)

This guide explains how to solve the exponential inequality 800(1.08)^n < 760(1.02)^(4n) by using logarithmic properties to simplify and solve for n. Covering concepts in inequalities, exponential functions, and logarithms, this solution is ideal for students in Grades 10-12.

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