Question

Помогите с логарифмами пожалуйста Дам 80 баллов

Answer 1

Ответ:

1.

$а) log_{6}(36) = 2$

$б) log_{2}( \frac{1}{64} ) = log_{2}( {2}^{ - 6} ) = - 6$

$в) log_{ \sqrt{5} }(5) = 2$

$г) log_{6}(1) = 0$

$д) log_{64}(8) = \frac{1}{2} = 0.5$

$е) log_{ 3 }(81) = 4$

$ж) log_{15}(15) = 1$

$з)lg(100000) = log_{10}( {10}^{5} ) = 5$

2.

$4 log_{2}(32) + log_{ \frac{1}{4} }(16) + log_{5}(125) + log_{4}(64) = 4 \times 5 - 2 + 3 + 3 = 20 + 4 = 24$

3.

$а) {8}^{ log_{8}(3) } = 3$

$б) {12}^{3 log_{12}(2) } = {12}^{ log_{12}( {2}^{3} ) } = {2}^{3} = 8$

$в) {4}^{ log_{4}(7) } + log_{ \frac{1}{2} }(16) = 7 - 4 = 3$

4.

$а) log_{ \sqrt{3} }( log_{ \frac{1}{5} }( \frac{1}{125} ) ) = log_{ \sqrt{3} }(3) = 2$

$б) log_{8}( log_{4}( log_{5}(625) ) ) = log_{8}( log_{4}(4) ) = log_{8}(1) = 0$

$в) {36}^{ log_{6}(3) } = {6}^{2 log_{6}(3) } = {6}^{ log_{6}( {3}^{2} ) } = {3}^{2} = 9$

$г) {5}^{ log_{125}(27) } = {5}^{ log_{ {5}^{3} }(27) } = {5}^{ \frac{1}{3} log_{5}(27) } = {5}^{ log_{5}( {27}^{ \frac{1}{3} } ) } = {27}^{ \frac{1}{3} } = 3$

$д) {16}^{ log_{4}(100) - log_{2}(5) } = {16}^{ log_{ {2}^{2} }(100) - log_{2}(5) } = {16}^{ \frac{1}{2} log_{2}(100) - log_{2}(5) } = {16}^{ log_{2}( \sqrt{100} ) - log_{2}(5) } = {16}^{ log_{2}( \frac{10}{5} ) } = {16}^{ log_{2}(2) } = 16$

$ж) log_{6}(25) \times log_{5}(216) = log_{6}( {5}^{2} ) \times log_{5}( {6}^{3} ) = 2 \times 3 \times log_{6}(5) \times log_{5}(6) = 2 \times 3 = 6$

5.

$а) log_{3}(x) = log_{ \frac{1}{3} }(5) \\ log_{3}(x) = log_{ {3}^{ - 1} }(5) \\ log_{3}(x) = - log_{3}(5) \\ log_{3}(x) = log_{3}( {5}^{ - 1} ) \\ x = {5}^{ - 1} \\ x = 0.2$

6.

$3) \frac{2 log_{4}(5) + log_{4}(9) }{ log_{4}(45) - log_{4}(3) } = \frac{ log_{4}( {5}^{2} \times 9 ) }{ log_{4}( \frac{45}{3} ) } = \frac{ log_{4}(225) }{ log_{4}(15) } = log_{15}(225) = 2$

7.

$а) {9}^{x} = 81 \\ x = 2$

$б) {3}^{x} = 11 \\ x = log_{3}(11)$

$в) {7}^{x} = 2 \\ x = log_{7}(2)$