Question

Wiedząc, że log4 3 = a oblicz log36 81.

Answer 1

Verified answer

Odpowiedź:

[tex]\huge\boxed {~~log_{4}3=a~~\Rightarrow ~~log_{36}81=\dfrac{4a}{2a+1} ~~}[/tex]

Szczegółowe wyjaśnienie:

Korzystamy ze wzorów:

• [tex]log_{a}a=1,~~zal.~~a > 0,~a\neq 1[/tex]
• [tex]log_{a}b=\dfrac{log_{c}b}{log_{c}a} ,~~zal.~~a > 0,~~a\neq 1,~~b > 0,~~c > 0,~~c\neq 1[/tex]
• [tex]log_{a}b+log_{a}c=log_{a}(b\cdot c),~~a > 0,~~a\neq 1,~~b > 0,~~c > 0[/tex]
• [tex]n\cdot log_{a}b=log_{a}b^{n},~zal.~~a > 0,~~a\neq 1,~~b > 0[/tex]

Rozwiązujemy :

[tex]log_{4}3=a\\\\log_{36}81=\dfrac{log_{4}81}{log_{4}36} =\dfrac{log_{4}3^{4}}{log_{4}(9\cdot4)} =\dfrac{4\cdot log_{4}3}{log_{4}9+log_{4}4} =\dfrac{4\cdot log_{4}3}{log_{4}3^{2}+log_{4}4} =\\\\=\dfrac{4\cdot log_{4}3}{2\cdot log_{4}3+log_{4}4} =\dfrac{4a}{2a+1}[/tex]