Question

NA TERAZ!
W razie problemu z odczytaniem ->
6. (Logarytm√3 z 3√3)⁴
7. Logarytm8 z 16 + 1

Answer 1

Odpowiedź:

6.

[tex]( log_{\sqrt{3}} 3\sqrt{3} )^4 = 3^4 - 81[/tex]

bo  ([tex]\sqrt{3} )^3 = 3\sqrt{3}[/tex]

7.

[tex]log_8 16 + 1 = log_{2^3} 2^4 + 1 = \frac{1}{3} *4 log_2 2 + 1 = \frac{4}{3} + \frac{3}{3}[/tex] [tex]= \frac{7}{3}[/tex]

8.

c = [tex]log_3 2[/tex]  ⇔  [tex]3^c = 2[/tex]

Szczegółowe wyjaśnienie:

Do  7.

[tex]log_{a^\alpha } x = \frac{1}{\alpha } log_ a x[/tex]

[tex]log_a x^n = n* log_a x[/tex]

Answer 2

Odpowiedź:

[tex]\huge\boxed{6. \ D. \ 81}[/tex]

[tex]\huge\boxed{7. \ C. \ \frac{7}{3}}[/tex]

[tex]\huge\boxed{8. \ C. \ 3^{c} = 2}[/tex]

Szczegółowe wyjaśnienie:

Z definicji logarytmu:

[tex]log_{a}b = c \ \ \ to \ \ \ a^{c} = b[/tex]

6.

[tex]log_{\sqrt{3}}3\sqrt{3} =x\\\\(\sqrt{3})^{x} = 3\sqrt{3}\\\\(3^{\frac{1}{2}})^{x} = 3^1\cdot3^{\frac{1}{2}}\\\\3^\frac{x}{2}}=3^{\frac{3}{2}}\\\\\frac{x}{2} = \frac{3}{2} \ \ \ |\cdot2\\\\x = 3\\\\log_{\sqrt{3}}3\sqrt{3} = 3\\\\(log_{\sqrt{3}}3\sqrt{3})^{4} = 3^{4} =\boxed{ 81}\\\\\underline{Odp. \ D.}[/tex]

7.

[tex]log_{8} 16 = x\\\\8^{x} = 16\\\\(2^{3})^{x} = 2^{4}\\\\2^{3x} = 2^{4}\\\\3x=4 \ \ \ /:3\\\\x = \frac{4}{3}\\\\log_{8}16+1 = \frac{4}{3}+1 = \frac{4}{3}+\frac{3}{3} =\boxed{ \frac{7}{3}}\\\\\underline{Odp. \ C.}[/tex]

8.

[tex]c = log_{3}2\\\\\boxed{3^{c} = 2}\\\\\underline{Odp. \ C.}[/tex]