https://file.aiquickdraw.com/mathbot/file/1729897975768yk9a7s4s.jpg

To solve for \( x \) to the nearest tenth, let's go through the equation provided:

\[
5.2 = 620 \times (0.5)^{\frac{x}{10}}
\]

### Steps to Solve

1. **Isolate** \( (0.5)^{\frac{x}{10}} \):
   \[
   (0.5)^{\frac{x}{10}} = \frac{5.2}{620}
   \]

2. **Simplify** the right side:
   \[
   (0.5)^{\frac{x}{10}} = 0.008387
   \]

3. **Take the logarithm** of both sides to solve for \( x \). We’ll use the natural logarithm:
   \[
   \ln \left( (0.5)^{\frac{x}{10}} \right) = \ln(0.008387)
   \]

4. **Apply the power rule** of logarithms on the left side:
   \[
   \frac{x}{10} \cdot \ln(0.5) = \ln(0.008387)
   \]

5. **Solve for \( x \)** by isolating it:
   \[
   x = \frac{10 \cdot \ln(0.008387)}{\ln(0.5)}
   \]

6. **Calculate** this expression to find \( x \).

Let me calculate this for you.The value of \( x \) to the nearest tenth is approximately:

\[
x \approx 69.0
\]

Would you like further details on this solution, or do you have any questions?

### Related Questions

1. How would the solution change if the base in the exponent was different from 0.5?
2. What steps would be different if the equation was instead \( 5.2 = 620(0.4)^{\frac{x}{10}} \)?
3. How do we approach solving exponential equations with different constants on both sides?
4. What are alternative methods to solve for \( x \) without logarithms?
5. How does rounding affect the precision of the solution?

### Tip
When dealing with equations involving exponents, logarithms are a powerful tool for isolating variables in the exponent.

Solve for x to the nearest 10th.

5.2 = 620(0.5)^(x/10)

Learn how to solve exponential equations using logarithms with this step-by-step solution to finding x in 5.2 = 620(0.5)^(x/10). Covers key concepts in exponential and logarithmic functions, suitable for high school students in Grades 10-12.

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