question\nsolve for $x$ to the nearest 10th.\n$3.1 = 600(0.5)^{\\frac{x}{7}}$

Explanation:

Step1: Isolate the exponential term

Divide both sides by 600. $$\frac{3.1}{600} = (0.5)^{\frac{x}{7}}$$ $$0.0051667 \approx (0.5)^{\frac{x}{7}}$$

Step2: Take natural log of both sides

Use logarithm power rule to bring down exponent. $$\ln(0.0051667) = \frac{x}{7} \ln(0.5)$$

Step3: Calculate log values

Compute the natural logarithms. $$-5.263 \approx \frac{x}{7} (-0.6931)$$

Step4: Solve for x

Isolate x by multiplying both sides by 7 and dividing by $\ln(0.5)$. $$x \approx \frac{-5.263 \times 7}{-0.6931}$$ $$x \approx \frac{-36.841}{-0.6931}$$ $$x \approx 53.15$$

Step5: Round to nearest tenth

Round the result to one decimal place. $$x \approx 53.2$$

Answer:

$x \approx 53.2$