1. **State the problem:** We need to find how many hours it takes for the moisture content to reduce from 42% to 14% if it decreases by 7% per hour. 2. **Formula and explanation:** The moisture content decreases by 7% each hour, meaning it retains 93% (100% - 7%) of the previous hour's moisture. This is an exponential decay problem modeled by: $$ M = M_0 \times (0.93)^t $$ where $M$ is the moisture after $t$ hours, $M_0$ is the initial moisture, and $0.93$ is the decay factor per hour. 3. **Set up the equation:** We want $M = 14$ and $M_0 = 42$, so: $$ 14 = 42 \times (0.93)^t $$ 4. **Isolate the exponential term:** $$ \frac{14}{42} = (0.93)^t $$ Simplify the fraction: $$ \frac{14}{42} = \cancel{\frac{14}{14}} \frac{1}{\cancel{3}} = \frac{1}{3} $$ So: $$ \frac{1}{3} = (0.93)^t $$ 5. **Take the natural logarithm of both sides:** $$ \ln\left(\frac{1}{3}\right) = \ln\left((0.93)^t\right) $$ Using the logarithm power rule: $$ \ln\left(\frac{1}{3}\right) = t \ln(0.93) $$ 6. **Solve for $t$:** $$ t = \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.93)} $$ 7. **Calculate the values:** $$ \ln\left(\frac{1}{3}\right) = \ln(1) - \ln(3) = 0 - 1.0986 = -1.0986 $$ $$ \ln(0.93) = -0.07257 $$ So: $$ t = \frac{-1.0986}{-0.07257} = 15.14 $$ 8. **Interpretation:** It will take approximately 15.14 hours for the moisture content to reduce from 42% to 14%. **Final answer:** $$ \boxed{15.14 \text{ hours}} $$