Задание 4.22

Question

4.22. Вычислите:
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1) $2^{4 \log _{2} 3-1}\qquad$
5) $12^{\log _{144} 4+\log _{12} 5}$
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2) $\left(\frac{1}{5}ight)^{\log _{25} 9+2}\qquad$
6) $1000^{\frac{1}{2} \lg 25-3 \lg 2}$;
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3) $8^{1-\frac{1}{3} \log _{2} 12}\qquad $
7) $\log _{13}\left(100^{\frac{1}{\log _{7} 10}}+2^{\log _{2} 15+3}ight)$;
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4) $6^{\frac{1}{2} \log _{6} 9-\log _{\frac{1}{6}} 3}\qquad$
8) $5^{\log _{5} 4 \cdot \log _{2} 3}$

Gimli · Accepted Answer

$$
\begin{array} 
	ext { 1) } 2^{4 \log _{2} 3-1}=\frac{\left(2^{\log _{2} 3}ight)^{4}}{2}=\frac{3^{4}}{2}=\frac{81}{2}=40,5 \
\
	ext { 2) }\left(\frac{1}{5}ight)^{\log _{25} 9+2}=\left(\frac{1}{5}ight)^{\log _{25} 9} \cdot \frac{1}{25}=\left(\frac{1}{5}ight)^{2 \log _{25} 3} \cdot \frac{1}{25}=\left(\frac{1}{25}ight)^{\log _{25} 3} \cdot \frac{1}{25}=\left(25^{ \log _{25} 3}ight)^{-1} \cdot \frac{1}{25}=\frac{1}{3} \cdot \frac{1}{25}=\frac{1}{75} \
\
\
	ext { 3) } 8^{1-\frac{1}{3} \log _{2} 12}=\left(2^{3}ight) ^{1-\frac{1}{3} \log _{2} 12}=2^{3-\log _{2} 12}=\frac{8}{2^{\log _{2} 12}}=\frac{2}{3}\
\
	ext { 4) } 6^{\frac{1}{2} \log _{6} 9-\log _{\frac{1}{6}} 3}=\frac{6^{\frac{1}{2} \log _{6} 9}}{6^{\log _{\frac{1}{6}} 3}}=\frac{6^{\log _{6} 3}}{6^{\log _\frac{1}{6} 3}}=\frac{3}{\left(\left(\frac{1}{6}ight)^{ \log _\frac{1}{6} 3}ight)^{-1}}=\frac{3}{\frac{1}{3}}=9\
\
	ext { 5) } 12^{\log _{144} 4+\log _{12} 5}=12^{\frac{1}{2} \log _{12} 4} \cdot 12^{\log _{12} 5}=\left(12^{\log _{12} 4}ight)^{\frac{1}{2}} \cdot 5=4^{\frac{1}{2}} \cdot 5=2 \cdot 5=10 	ext {. }\
\
	ext { 6) } 1000^{\frac{1}{2} \lg 25-3 \lg 2}=\frac{1000^{\frac{1}{2} \lg 25}}{1000^{3 \lg 2}}=\frac{\left(\left(10^{\lg 25}ight)^3ight)^{\frac{1}{2}}}{\left(10^9ight)^{\lg 2}}=\frac{25^{\frac{3}{2}}}{\left(10^{\lg  2}ight)^{9}}=\frac{\sqrt{5^{6}}}{2^{9}}=\frac{5^{3}}{2^{9}}=\frac{125}{512}\
\
	ext { 7) } \log _{13}\left(100^{\frac{1}{\log _{7} 10}}+2^{\log _{2} 15+3}ight)=\log _{13}\left(100^{\lg  7}+15 \cdot 2^{3}ight)=\log _{13}(49+120)=\log _{13} 169=2\
\
\
	ext { 8) } 5^{\log _{5} 4 \cdot \log _{2} 3}=\left(5^{\log _{5} 4}ight) ^{\log _{2} 3}=4^{\log _{2} 3}=\left(2^{\log _{2} 3}ight)^2=3^{2}=9\

\end{array}
$$

Задание 4.22

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Задание 4.22

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