Вычислите:

a) $$log_{5}\frac{1}{25} + log_{\sqrt{3}}27$$;

• $$log_{5}\frac{1}{25} = log_{5}5^{-2} = -2log_{5}5 = -2$$
• $$log_{\sqrt{3}}27 = log_{3^{1/2}}3^{3} = \frac{3}{1/2}log_{3}3 = 6$$

$$log_{5}\frac{1}{25} + log_{\sqrt{3}}27 = -2 + 6 = 4$$

Ответ: 4

б) $$log_{1.5}log_{4}8$$;

• $$log_{4}8 = log_{2^{2}}2^{3} = \frac{3}{2}log_{2}2 = \frac{3}{2} = 1.5$$

$$log_{1.5}log_{4}8 = log_{1.5}1.5 = 1$$

Ответ: 1

в) $$4^{log_{2}3+0.5log_{2}9}$$;

• $$4^{log_{2}3+0.5log_{2}9} = 4^{log_{2}3+log_{2}9^{0.5}} = 4^{log_{2}3+log_{2}3} = 4^{2log_{2}3} = 4^{log_{2}3^{2}} = 4^{log_{2}9} = (2^{2})^{log_{2}9} = 2^{2log_{2}9} = 2^{log_{2}9^{2}} = 2^{log_{2}81} = 81$$

Ответ: 81

г) $$10^{lg\frac{1}{5}-lg2}$$

• $$10^{lg\frac{1}{5}-lg2} = 10^{lg\frac{1}{5} / 2} = 10^{lg\frac{1}{10}} = 10^{-1} = \frac{1}{10} = 0.1$$

Ответ: 0.1