1.5 Logarytmy #

Wzory

$ log_{a}xy = log_{a}x + log_{a}y $

$ log_{a}\frac{x}{y} = log_{a}x - log_{a}y $

$ log_{a}x = \frac{log_{b}x}{log_{b}a} $

$ log_{a}x^b = b\cdot log_{a}x $

$ log_{a}a = 1 $

$ log_{a}a^x = x\cdot log_{a}a = x $

$ log_{a}1 = 0 $

$ a^{log_{a}x} = log_{a}a^x = x $

Zadania #

1.5.2 Oblicz logarytmy #

a) $ log_{3}27 = $

b) $ log_{2}32 = $

c) $ log_{4}2 = $

d) $ log1000 = $

e) $ log_{2}1 = $

f) $ log_{2}{\frac{1}{8}} = $

Rozwiązanie

a) $ log_{3}27 = 3 $, bo $3^3 = 27 $

b) $ log_{2}32 = 5 $, bo $2^5 = 32 $

c) $ log_{4}2 = \frac{1}{2} $, bo $4^{\frac{1}{2}} = \sqrt{4} = 2 $

d) $ log1000 = 3 $, bo $10^3 = 1000$

e) $ log_{2}1 = 0$, bo $ 2^0 = 1$

f) $ log_{2}{\frac{1}{8}} = -3$, bo $ 2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

1.5.3 Oblicz logarytmy #

a) $ log_{2}\frac{1}{1024} = $

b) $ log_{\frac{2}{3}}{\frac{81}{16}} = $

c) $log_{0.16}0.064 = $

d) $ log_{0.125}0.5 = $

e) $ log_{3}25 \cdot log_{5}27 = $

f) $ \frac{3^{log_{2}40}}{ 3^{log_{2}5}} = $

g) $ log_{\sqrt[3]{3}} 27 = $

h) $ log_{\frac{1}{5}} 25\sqrt{5} = $

i) $ log_{2\sqrt{2}} 4\sqrt{8} = $

Rozwiązanie

a) $ log_{2}\frac{1}{1024} = log_{2}1024^{-1} = -log_{2}2^{10} = -10log_{2}2 = -10 $

b) $ log_{\frac{2}{3}}{\frac{81}{16}} = log_{\frac{2}{3}}{(\frac{16}{81}})^{-1} = -log_{\frac{2}{3}}{(\frac{2}{3}})^{4} = -4log_{\frac{2}{3}}{(\frac{2}{3}}) = -4 $

c) $log_{0.16}0.064 = log_{0.16}{(0.4)}^3 = 3log_{0.16}\sqrt{0.16}= 3log_{0.16}{(0.16)}^{\frac{1}{2}} = $ $ = \frac{3}{2}log_{0.16}0.16 = \frac{3}{2} $

d) $ log_{0.125}0.5 = log_{0.125}(0.125)^{\frac{1}{3}} = \frac{1}{3}log_{0.125}0.125 = \frac{1}{3}$

e) $ log_{3}25 \cdot log_{5}27 = \frac{log_{5}25}{log_{5}3} \cdot log_{5}27 = \frac{2}{log_{5}3} \cdot log_{5}3^3 = \frac{2}{log_{5}3} \cdot 3\cdot log_{5}3 = 6$

f) $ \frac{3^{log_{2}40}}{ 3^{log_{2}5}} = 3^{log_{2}40 - log_{2}5} = 3^{log_{2}\frac{40}{5}} = 3^{log_{2}8} = 3^3 = 27 $

g) $ log_{\sqrt[3]{3}} 27 = log_{3^{\frac{1}{3}}}3^3 = log_{3^{\frac{1}{3}}}(3^{\frac{1}{3}})^9 = 9log_{3^{\frac{1}{3}}}3^{\frac{1}{3}} = 9$

h) $ log_{\frac{1}{5}} 25\sqrt{5} = log_{\frac{1}{5}}5^2 \cdot 5^{\frac{1}{2}} = log_{\frac{1}{5}}5^{\frac{5}{2}} = \frac{5}{2}log_{\frac{1}{5}}5 = \frac{5}{2}log_{\frac{1}{5}}{(\frac{1}{5}})^{-1} = -\frac{5}{2}$

i) $ log_{2\sqrt{2}} 4\sqrt{8} = \frac{log_{2}4\sqrt{8}}{log_2{2\sqrt{2}}} = \frac{log_{2}{2^2 \cdot 2^{\frac{3}{2}}}}{log_{2}2 \cdot 2^{\frac{1}{2}}} = \frac{log_{2}2^{\frac{7}{2}}}{log_{2}2^{\frac{3}{2}}} = \frac{\frac{7}{2}log_{2}2}{\frac{3}{2}log_{2}2} = \frac{7}{3} = 2\frac{1}{3}$