1. **State the problem:** We want to find the time $h$ it takes for the bacteria colonies to grow from 10 to 15,080 given the growth rate is 5% per hour. 2. **Write the formula:** The number of bacteria colonies after $h$ hours is given by $$N(h) = B \cdot r^{2h}$$ where $B$ is the initial amount, $r$ is the growth rate factor, and $h$ is the time in hours. 3. **Identify given values:** - Initial colonies $B = 10$ - Growth rate $5\% = 0.05$, so $r = 1 + 0.05 = 1.05$ - Final colonies $N(h) = 15,080$ 4. **Set up the equation:** $$15,080 = 10 \cdot (1.05)^{2h}$$ 5. **Isolate the exponential term:** $$\frac{15,080}{10} = (1.05)^{2h}$$ $$1,508 = (1.05)^{2h}$$ 6. **Take the natural logarithm of both sides:** $$\ln(1,508) = \ln\left((1.05)^{2h}\right)$$ 7. **Use logarithm power rule:** $$\ln(1,508) = 2h \cdot \ln(1.05)$$ 8. **Solve for $h$:** $$h = \frac{\ln(1,508)}{2 \cdot \ln(1.05)}$$ 9. **Calculate values:** $$\ln(1,508) \approx 7.318$$ $$\ln(1.05) \approx 0.04879$$ 10. **Final calculation:** $$h = \frac{7.318}{2 \times 0.04879} = \frac{7.318}{0.09758} \approx 75.0$$ **Answer:** It will take approximately **75 hours** for the bacteria colonies to grow to 15,080.