question\nsolve for $x$ to the nearest 10th.\n$0.3 = 1.03^{\\frac{x}{3}+5} - 1$

Explanation:

Step1: Isolate the exponential term

Add 1 to both sides. $$0.3 + 1 = 1.03^{\frac{x}{3}+5}$$ $$1.3 = 1.03^{\frac{x}{3}+5}$$

Step2: Convert to logarithmic form

Take $\log_{10}$ of both sides. $$\log(1.3) = \left(\frac{x}{3}+5\right)\log(1.03)$$

Step3: Solve for the linear term

Divide both sides by $\log(1.03)$. $$\frac{\log(1.3)}{\log(1.03)} = \frac{x}{3}+5$$ Calculate the left-hand side: $\frac{\log(1.3)}{\log(1.03)} \approx 8.849$

Step4: Isolate the $x$-term

Subtract 5 from both sides. $$8.849 - 5 = \frac{x}{3}$$ $$3.849 = \frac{x}{3}$$

Step5: Solve for $x$

Multiply both sides by 3. $$x = 3.849 \times 3$$ $$x \approx 11.547$$

Step6: Round to nearest 10th

Round $11.547$ to one decimal place. $$x \approx 11.5$$

Answer:

$11.5$