1. **State the problem:** Solve for $x$ in the equation $100 = 85 \times 1.005^x$. 2. **Write the formula and explain:** We want to isolate $x$. The equation involves an exponential term $1.005^x$. To solve for $x$, we will first divide both sides by 85, then use logarithms to bring down the exponent. 3. **Divide both sides by 85:** $$\frac{100}{85} = \frac{85 \times 1.005^x}{85}$$ $$\Rightarrow \frac{100}{85} = \cancel{\frac{85}{85}} \times 1.005^x$$ $$\Rightarrow \frac{100}{85} = 1.005^x$$ 4. **Take the natural logarithm (ln) of both sides:** $$\ln\left(\frac{100}{85}\right) = \ln\left(1.005^x\right)$$ 5. **Use the logarithm power rule:** $$\ln\left(1.005^x\right) = x \ln(1.005)$$ 6. **Rewrite the equation:** $$\ln\left(\frac{100}{85}\right) = x \ln(1.005)$$ 7. **Solve for $x$ by dividing both sides by $\ln(1.005)$:** $$x = \frac{\ln\left(\frac{100}{85}\right)}{\ln(1.005)}$$ 8. **Calculate the values:** $$\frac{100}{85} \approx 1.17647$$ $$\ln(1.17647) \approx 0.16252$$ $$\ln(1.005) \approx 0.0049875$$ 9. **Final calculation:** $$x \approx \frac{0.16252}{0.0049875} \approx 32.59$$ **Answer:** $x \approx 32.59$