Question

помогите подробно решить логарифмы

Answer 1

Ответ:

$1) {(0.5)}^{ log_{ \sqrt{2} }(3) } = {2}^{ - 1 \times log_{ {(2)}^{ \frac{1}{2} } }(3) } = {2}^{ - 1 \times 2 log_{2}(3) } = {2}^{ log_{2}( {3}^{ - 2} ) } = {3}^{ - 2} = \frac{1}{9}$

$2) { \sqrt[3]{2} }^{ log_{4}(3) } = {2}^{ \frac{1}{3} log_{4}(3) } = {2}^{ \frac{1}{3} log_{ {2}^{2} }(3) } = {2}^{ \frac{1}{3} \times \frac{1}{2} log_{2}(3) } = {2}^{ \frac{1}{6} log_{2}(3) } = {2}^{ log_{2}( {3}^{ \frac{1}{6} } ) } = \sqrt[6]{3}$

$3) {(0.4)}^{2 log_{2.5}(2) } = {( \frac{2}{5}) }^{2 log_{ \frac{5}{2} }(2) } = {( \frac{5}{2}) }^{ -2 log_{ \frac{5}{2} }(2) } = {( \frac{5}{2}) }^{ log_{ \frac{5}{2} }( {2}^{ - 2} ) } = {2}^{ - 2} = \frac{1}{4}$

$4) {2}^{3 log_{2}(5) + log_{2}(3) } = {2}^{ log_{2}( {5}^{3} ) } \times {2}^{ log_{2}(3) } = {5}^{3} \times 3 = 125 \times 3 = 375$