Question

Log de x na base 3 * log de x na base 9 * log de x na base 27 * log de x na base 81 = 2/3

Answer 1

Resposta:

x=9

Explicação passo-a-passo:

log₃x.log₉x.log₂₇x.log₈₁x=2/3

Mudança de base:

log₉x=log₃x/log₃9=log₃x/log₃3²=log₃x/2log₃3=log₃x/2

log₂₇x=log₃x/log₃27=log₃x/log₃3³=log₃x/3log₃3=log₃x/3

log₈₁x=log₃x/log₃81=log₃x/log₃3⁴=log₃x/4log₃3=log₃x/4

log₃x.log₃x/2.log₃x/3.log₃x/4=2/3

(log₃x)⁴=24.2/3

(log₃x)⁴=16

(log₃x)⁴=2⁴

log₃x=2

3²=x

x=9

Answer 2

Resposta:

Explicação passo-a-passo:

$(log_3x).(log_9x).(log_{27}x).(log_{81}x)=\dfrac{2}{3}$

mudando para a base 3

$(log_3x).(\dfrac{log_3x}{log_39}).(\dfrac{log_3x}{log_327}).(\dfrac{log_3x}{log_381})=\dfrac{2}{3}\\\\\\(log_3x).(\dfrac{log_3x}{2}).(\dfrac{log_3x}{3}).(\dfrac{log_3x}{4})=\dfrac{2}{3}\\\\\\\dfrac{(log_3x)^{4}}{24}=\dfrac{2}{3}\\\\\\(log_3x)^{4}=\dfrac{24.2}{3}\\\\\\(log_3x)^{4}=16\\\\\\(log_3x)^{4}=2^{4}\\ \\log_3x=2\\\\x=3^{2}\\ \\\boxed{\boxed{x=9}}$