Question

помогите
найти
$log_{ {a}^{2} b}(ab)$
если
$log_{ \frac{b}{a} }( \frac{ {a}^{2} }{b} ) = - \frac{1}{2}$
​

Answer 1

Ответ: 4/5

Объяснение:

Первый способ.

log(b/a; a^2/b) = log(b/a; a/b) + log(b/a; a)  = -1/2

-1 +  log(b/a; a)  =-1/2

log(b/a; a)  = 1/2

log(a; b/a) = 2

log(a; 1/a) +log(a;b) = 2

-1 + log(a; b) = 2

log(a; b) = 3

log(ab; a^2*b) = log(ab; ab) +log(ab; a) =

=1+ 1/log(a; ab) = 1+1/( log(a;a) +log(a;b) = 1+ 1/4 = 5/4

log(a^2*b; ab) = 1/(5/4) = 4/5

Второй способ.

log(b/a; a^2/b) = -1/2

(b/a)^(-1/2) = a^2/b

a/b =a^4/b^2

a^3=b

log(a^2*b; ab)  = log(a^5; a^4) = 4/5

Answer 2

Объяснение:

Найти $log_{a^2b}(ab)$, если $log_{\frac{b}{a}} (\frac{a^2}{b})=-\frac{1}{2}$

Рассмотрим $log_{\frac{b}{a}} (\frac{a^2}{b})=-\frac{1}{2}$

$\frac{a^2}{b}=(\frac{b}{a})^{-\frac{1}{2} }\\\frac{a^2}{b}=(\frac{a}{b})^{\frac{1}{2} }\\(\frac{a^2}{b})^2=((\frac{a}{b})^{\frac{1}{2} })^2\\\frac{a^4}{b^2} =\frac{a}{b}\\\frac{b^2}{b}=\frac{a^4}{a} \\b=a^3.\Rightarrow$

$log_{a^2b}(ab)=log_{a^2*a^3}(a*a^3)=log_{a^5}(a^4)=\frac{1}{5} *log_a(a^4)=\frac{4}{5} *log_aa=\frac{4}{5}.$

Ответ: $log_{a^2b}(ab)=\frac{4}{5}.$