Question

Известно, что:
$log_ax=\alpha, ~~log_bx=\beta , ~~log_cx=\gamma, ~~log_dx=\delta, ~~ x\neq 1$
Найти:
$log_{abcd}x$

Answer 1

$log_{abcd}x=\frac{1}{log_{x}abcd}=\frac{1}{log_{x}a+log_{x}b+log_{x}c+log_{x}d}=\\\\=\frac{1}{\frac{1}{log_{a} x}+\frac{1}{log_{b} x} +\frac{1}{log_{c} x}+\frac{1}{log_{d} x}}=\frac{1}{\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}} =\frac{\alpha \beta \gamma\delta }{\beta \gamma\delta+\alpha \gamma \delta+\alpha \beta \delta+\alpha \beta \gamma }$