a)
log_232=log_22^5=5
log_{\frac{1}{7}}7=log_{\frac{1}{7}}(\frac{1}{7})^{-1}=-1
log_42=log_4\sqrt4=log_44^{\frac{1}{2}}=\frac{1}{2}
log_{0,3}0,027=log_{0,3}(0,3)^3=3
lub
log_{0,3}0,027=log_{0,3}\frac{27}{1000}=log_{0,3}(\frac{3}{10})^3=log_{0,3}0,3^3=3
log_{0,1}100=log_{0,1}10^2=log_{0,1}(\frac{1}{10})^{-2}=log_{0,1}0,1^{-2}=-2
b)
log_55=1 gdyż 5^1=5
log_71=0 gdyż 7^0=1
log0,1=log_{10}\frac{1}{10}=log_{10}10^{-1}=-1
log\sqrt{10}=log_{10}10^{\frac{1}{2}}=\frac{1}{2}
c)
log_{\frac{1}{2}}4=log^_{\frac{1}{2}}2^2=log_{\frac{1}{2}}(\frac{1}{2})^{-2}=-2
log_9\frac{1}{3}=log_93^{-1}=log_9(\sqrt9)^{-1}=log_9((9^{\frac{1}{2}})^{-1}=log_99^{-\frac{1}{2}}=-\frac{1}{2}
log_5\frac{1}{125}=log_5125^{-1}=log_5(5^3)^{-1}=log_55^{-3}=-3
log_{\sqrt5}5=log_{\sqrt5}(\sqrt5)^2=2
d)
log_55^3=3
log_88^{\frac{1}{3}}=\frac{1}{3}
log_44^{-\frac{2}{7}}=-\frac{2}{7}
log_{\frac{1}{2}}2^{-4}=log_{\frac{1}{2}}(\frac{1}{2})^4=4
log_7(7^{-3})^{\frac{1}{5}}=log_77^{-\frac{3}{5}}=-\frac{3}{5}