経済学で出る数学

ワークブックでじっくり攻める：問 3.31

1. $\log_{2}{16}=\log_{2}{2^4}=4\log_{2}{2}=4$


2. $\log_{2}{\sqrt{2}}=\log_{2}{2^{\frac{1}{2}}}=\dfrac{1}{2}\log_{2}{2}=\dfrac{1}{2}$


3. $\log_{10}{\sqrt[3]{100}}=\log_{10}{10^{\frac{2}{3}}}=\dfrac{2}{3}\log_{10}{10}=\dfrac{2}{3}$


4. $\log_{6}{\dfrac{3}{2}}+\log_{6}{24}=\log_{6}{\left(\dfrac{3}{2}\times 24\right)}= \log_{6}{36}=\log_{6}{6^2}=2$


5. $\log_{3}{15}-\log_{3}{5}=\log_{3}{\dfrac{15}{5}}= \log_{3}{3}=1$


6. $\log_{3}{5}-\log_{3}{15}=\log_{3}{\dfrac{5}{15}}= \log_{3}{\dfrac{1}{3}}=\log_{3}{3^{-1}}=-1$


7. $\log_{5}{20}+\log_{5}{50}-3\log_{5}{2}=\log_{5}{\dfrac{20\times 50}{2^3}}= \log_{5}{125}=\log_{5}{5^{3}}=3$


8. $\log_{2}{\dfrac{2}{3}}+\log_{2}{6}-\log_{2}{4} =\log_{2}{\dfrac{2\times 6}{3\times 4}}= \log_{2}{\dfrac{12}{12}}=\log_{2}{1}=0$