Задание №4. Найдите значение логарифмической функции y = log₂x указанных точках: a) x₁ = 4, x₂ = 8, x₃ = 16; б) x₁ = 2/√8, x₂ = 4/√2;

Решение:

Задан логарифм \( y = \log_2 x \).

а)

При \( x_1 = 4 \):

\[ y_1 = \log_2 4 = \log_2 2^2 = 2 \]

При \( x_2 = 8 \):

\[ y_2 = \log_2 8 = \log_2 2^3 = 3 \]

При \( x_3 = 16 \):

\[ y_3 = \log_2 16 = \log_2 2^4 = 4 \]

б)

При \( x_1 = \frac{2}{\sqrt{8}} \):

\[ y_1 = \log_2 \left( \frac{2}{\sqrt{8}} \right) = \log_2 \left( \frac{2}{2^{3/2}} \right) = \log_2 \left( 2^{1 - 3/2} \right) = \log_2 \left( 2^{-1/2} \right) = -\frac{1}{2} \]

При \( x_2 = \frac{4}{\sqrt{2}} \):

\[ y_2 = \log_2 \left( \frac{4}{\sqrt{2}} \right) = \log_2 \left( \frac{2^2}{2^{1/2}} \right) = \log_2 \left( 2^{2 - 1/2} \right) = \log_2 \left( 2^{3/2} \right) = \frac{3}{2} \]

Ответ: а) y₁=2, y₂=3, y₃=4; б) y₁=-1/2, y₂=3/2.