Question

Se log4 512 = e log9 243 = y, então x + y vale:​

Answer 1

Explicação passo-a-passo:

Usando a definição de logaritmo:

$\log_{a}b=x\implies a^{x}=b$

Calculando o valor de x:

$\log_{4}512=x\\\\4^{x}=512\\\\(2^{2} )^{x}=2^{9} \\\\2^{2x}=2^{9}\\\\\text Igualando\ os\ expoentes:\\\\2x=9\\\\\boxed{x=\dfrac{9}{2}}$

Calculando o valor de y:

$\log_{9}243=y\\\\9^{y}=243\\\\(3^{2} )^{y}=3^{5} \\\\3^{2y}=3^{5}\\\\\text Igualando\ os\ expoentes:\\\\2y=5\\\\\boxed{y=\dfrac{5}{2}}$

Calculando x+y:

$x+y=\dfrac{9}{2} +\dfrac{5}{2} \\\\x+y=\dfrac{14}{2}\\\\\boxed{\boxed{x+y=7}}$