Question

Помогите решить логарифмы

Answer 1

$5^{1+log_5 3}=5cdot5^{log_5 3}=5cdot3=15, \ 10^{1-lg 2}=frac{10}{10^{lg2}}=frac{10}{2}=5, \ (frac{1}{7})^{1+log_{frac{1}{7}} 2}=frac{1}{7}cdot(frac{1}{7})^{log_{frac{1}{7}} 2}=frac{1}{7}cdot2=frac{2}{7}, \ 3^{2-log_3 18}=frac{3^2}{3^{log_3 18}}=frac{9}{18}=frac{1}{9};$

 

$4^{2log_2 3}=(2^2)^{2log_2 3}=2^{4log_2 3}=(2^{log_2 3})^4=3^4=81, \ 5^{-3log_5 frac{1}{2}}=(5^{log_5 frac{1}{2}})^{-3}=(frac{1}{2})^{-3}=2^3=8, \ (frac{1}{2})^{4log_{frac{1}{2}} 3}=((cc)^{log_{frac{1}{2}} 3})^4=3^4=81, \ 6^{-2log_6 5}=(6^{log_6 5})^{-2}=5^{-2}=frac{1}{5^2}=frac{1}{25};$

 

$log_3 (sqrt[5]{a^3b})^{frac{2}{3}}=log_3 ((a^3b)^frac{1}{5})^frac{2}{3}=log_3 (a^3b)^frac{2}{15}=log_3 ((a^3)^frac{2}{15}b^frac{2}{15})= \ =log_3 (a^frac{2}{5}b^frac{2}{15})=log_3 a^frac{2}{5}+log_3 b^frac{2}{15}=frac{2}{5} log_3 a + frac{2}{15} log_3 b; \ log_3 (9a^4sqrt[5]{b})=log_3 3^2 + log_3 a^4 + log_3 b^frac{1}{5}=2+4log_3 a + frac{1}{5}log_3 b.$

Answer 2

5^{1+log_5 3}=5cdot5^{log_5 3}=5cdot3=15, \ 10^{1-lg 2}=frac{10}{10^{lg2}}=frac{10}{2}=5, \ (frac{1}{7})^{1+log_{frac{1}{7}} 2}=frac{1}{7}cdot(frac{1}{7})^{log_{frac{1}{7}} 2}=frac{1}{7}cdot2=frac{2}{7}, \ 3^{2-log_3 18}=frac{3^2}{3^{log_3 18}}=frac{9}{18}=frac{1}{9};