Question

Urgent!! Dau coroana!!!
Numerele reale x si y verifica 2^x=5 si 5^y=2 . Calculati xy.

Answer 1

$\left\{ \begin{array}{c} 2^x = 5 \\ 5^y = 2 \end{array} \right \Rightarrow \left\{ \begin{array}{c} x = log_\big25\\ y = log_\big52 \end{array} \right \Rightarrow \left\{ \begin{array}{c} x = log_\big25\\ y = \dfrac{log_\big22}{log_\big25} \end{array} \right \Rightarrow \left\{ \begin{array}{c} x = log_\big25\\ y = \dfrac{1}{log_\big25} \end{array} \right \Rightarrow$
$\Rightarrow x\cdot y = log_\big25\cdot\dfrac{1}{log_\big25} \Rightarrow x\cdot y = \dfrac{log_\big25}{log_\big25}\Rightarrow \boxed{ x\cdot y = 1}$

$\ \Big( M-am folosit de formula de schimbare a bazei: log_\big{a}b= \dfrac{log_\big{c}b}{log_\big{c}a} \Big)$

Answer 2

[tex]\it 2^x =5 \ \ \ \ \ (1) \\\;\\ 5^y=2 \ \ \ \ \ (2) [/tex]


$\it 2^x=5 \Rightarrow (2^x)^y=5^y \stackrel{(2)}{\Longrightarrow} 2^{xy} = 2 \Rightarrow xy = 1$