Question

Rewrite the following in terms of ln 2, ln 3, and ln 5. Then use ln 2=0.693, ln 3=1.099, and ln 5=1.609.

1. ln 6
2. ln (10/3)
3. ln 30
4. ln 12
5. ln (2/5)
5. ln (5/6)

Answer 1

`\log_a (b * c)=\log_a b + \log_a c \\ \log_a ((b)/(c))=\log_a b-\log_a c \\ \log_a b^c=c \log_a b \\ \\ 1. \\ \ln 6=\ln (2 * 3)=\boxed{\ln 2 + \ln 3} \approx 0.693+1.099=\boxed{1.792} \\ \\ 2. \\ \ln ((10)/(3))=\ln ((2 * 5)/(3))=\boxed{\ln 2 + \ln 5 - \ln 3} \approx 0.693+1.609-1.099=\boxed{1.203} \\ \\ 3. \\ \ln 30=\ln (2 * 3 * 5)=\boxed{\ln 2+ \ln 3 + \ln 5} \approx 0.693+ 1.099 + 1.609=\\=\boxed{3.401}`


`4. \\ \ln 12=\ln (2^2 * 3)=\ln 2^2 + \ln 3=\boxed{2 \ln 2+ \ln 3} \approx 2 * 0.693+ 1.099= \\ =\boxed{2.485} \\ \\ 5. \\ \ln ((2)/(5))=\boxed{\ln 2 - \ln 5} \approx 0.693-1.609=\boxed{-0.916} \\ \\ 6. \\ \ln ((5)/(6))=\ln ((5)/(2 * 3))=\ln 5-(\ln 2+ \ln 3)=\boxed{\ln 5 - \ln 2 - \ln 3} \approx \\ \approx1.609-0.693-1.099=\boxed{-0.183}`