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Let's solve the equation \( 9 = 80(1.9)^x \) using logarithms.

### Steps:

1. Start by isolating the exponential term:
   \[
   \frac{9}{80} = (1.9)^x
   \]
   Simplifying the left side:
   \[
   0.1125 = (1.9)^x
   \]

2. Now, apply the natural logarithm (or any logarithm) to both sides:
   \[
   \ln(0.1125) = \ln((1.9)^x)
   \]

3. Using the logarithmic property \( \ln(a^b) = b\ln(a) \), we can simplify the right side:
   \[
   \ln(0.1125) = x \ln(1.9)
   \]

4. Solve for \( x \):
   \[
   x = \frac{\ln(0.1125)}{\ln(1.9)}
   \]

5. Calculate the values:
   \[
   \ln(0.1125) \approx -2.1848, \quad \ln(1.9) \approx 0.6419
   \]
   So,
   \[
   x = \frac{-2.1848}{0.6419} \approx -3.4
   \]

Thus, \( x \approx -3.4 \).

Would you like more details or clarifications on any step?

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Here are 5 related questions for further practice:
1. Solve \( 12 = 100(2.3)^x \) using logarithms.
2. What is the relationship between logarithms and exponents?
3. How do you simplify logarithmic expressions using properties of logarithms?
4. Solve \( 5^x = 20 \) using natural logarithms.
5. How do you apply logarithms to solve exponential growth problems?

**Tip:** When solving exponential equations, logarithms are extremely useful for "bringing down" exponents and simplifying the equation.

Solve the equation 9 = 80(1.9)^x for x using logs.

Learn how to solve the exponential equation 9 = 80(1.9)^x using logarithms. This problem covers core concepts such as applying logarithmic properties to isolate variables in an exponential equation. Suitable for students in Grades 10-12.

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