1. **Stating the problem:** We are given bounds for $\ln 2$ and $\ln 5$ and asked to estimate values of logarithmic expressions such as $\ln 4$, $\ln(2^5)$, $\ln\left(\frac{5}{2}\right)$, and $\ln\left(\frac{16}{25}\right)$. We will use properties of logarithms and the given bounds. 2. **Recall logarithm properties:** - $\ln(a^b) = b \ln a$ - $\ln\left(\frac{a}{b}\right) = \ln a - \ln b$ - $\ln 4 = \ln(2^2) = 2 \ln 2$ 3. **Calculate $\ln 4$:** Given $0.69 < \ln 2 < 0.70$, then $$\ln 4 = 2 \ln 2$$ So, $$2 \times 0.69 = 1.38 < \ln 4 < 2 \times 0.70 = 1.40$$ 4. **Calculate $\ln(2^5)$:** $$\ln(2^5) = 5 \ln 2$$ Using bounds, $$5 \times 0.69 = 3.45 < \ln(2^5) < 5 \times 0.70 = 3.50$$ 5. **Calculate $\ln\left(\frac{5}{2}\right)$:** $$\ln\left(\frac{5}{2}\right) = \ln 5 - \ln 2$$ Given $1.60 < \ln 5 < 1.61$ and $0.69 < \ln 2 < 0.70$, then $$1.60 - 0.70 = 0.90 < \ln\left(\frac{5}{2}\right) < 1.61 - 0.69 = 0.92$$ 6. **Calculate $\ln\left(\frac{16}{25}\right)$:** Rewrite numerator and denominator: $$16 = 2^4, \quad 25 = 5^2$$ So, $$\ln\left(\frac{16}{25}\right) = \ln(2^4) - \ln(5^2) = 4 \ln 2 - 2 \ln 5$$ Using bounds, $$4 \times 0.69 = 2.76 < 4 \ln 2 < 4 \times 0.70 = 2.80$$ $$2 \times 1.60 = 3.20 < 2 \ln 5 < 2 \times 1.61 = 3.22$$ Therefore, $$2.76 - 3.22 = -0.46 < \ln\left(\frac{16}{25}\right) < 2.80 - 3.20 = -0.40$$ **Final answers:** - $\ln 4 \in (1.38, 1.40)$ - $\ln(2^5) \in (3.45, 3.50)$ - $\ln\left(\frac{5}{2}\right) \in (0.90, 0.92)$ - $\ln\left(\frac{16}{25}\right) \in (-0.46, -0.40)$ These intervals give approximate ranges for the logarithms based on the given bounds for $\ln 2$ and $\ln 5$.