Question

Relembrando propriedades de logaritmo e resolvendo a equação logarítmica 3^log_3⁡7 -4^log_4⁡2 =5^((log_5⁡x-log_5⁡〖x^2 〗 ) ) será x=1/5 ??

Answer 1

Explicação passo-a-passo:

Lembre-se que:

• $\sf log_{b}~a-log_{b}~c=log_{b}~\Big(\dfrac{a}{c}\Big)$

• $\sf b^{log_{b}~a}=a$

Assim:

• $\sf 3^{log_{3}~7}=7$

• $\sf 4^{log_{4}~2}=2$

• $\sf log_{5}~x-log_{5}~x^2=log_{5}~\Big(\dfrac{x}{x^2}\Big)$

$\sf log_{5}~x-log_{5}~x^2=log_{5}~\Big(\dfrac{1}{x}\Big)$

• $\sf 5^{log_{5}~x-log_{5}~x^2}=5^{log_{5}~\Big(\frac{1}{x}\Big)}$

$\sf 5^{log_{5}~x-log_{5}~x^2}=\dfrac{1}{x}$

Logo:

$\sf 3^{log_{3}~7}-4^{log_{4}~2}=5^{log_{5}~x-log_{5}~x^2}$

$\sf 7-2=5^{log_{5}~\Big(\frac{x}{x^2}\Big)}$

$\sf 5=5^{log_{5}~\Big(\frac{1}{x}\Big)}$

$\sf 5=\dfrac{1}{x}$

$\sf 5x=1$

$\sf \red{x=\dfrac{1}{5}}$

Answer 2

Resposta:

$\textsf{Leia abaixo}$

Explicação passo-a-passo:

$\sf 3^{log_3\:7}-4^{log_4\:2} = 5^{log_5\:x - log_5\:x^2}$

$\sf 3^{log_3\:7}-4^{log_4\:2} = 5^{log_5\:\frac{x}{x^2}$

$\sf 3^{log_3\:7}-4^{log_4\:2} = 5^{log_5\:\frac{1}{x}$

$\sf 7 - 2 = \dfrac{1}{x}$

$\sf 5 = \dfrac{1}{x}$

$\boxed{\boxed{\sf x = \dfrac{1}{5}}}$