Question

Qual o valor do logaritmo base 9 e logaritmando três vezes a raiz quadrada de vinte e sete ?

Answer 1

$log_9(3.\sqrt{27})  = log_9(3.\sqrt{3^3}) =  log_9(3.3\sqrt{3}) = log_( 9.\sqrt{3}) = log_9(9)+log_9(\sqrt{3}) = 1 + log_9(3^{\frac{1}{2}}) = 1+\frac{1}{2}.log_9(3)$

Obs.: $log_9(3)=x \implies 9^x = 3 \implies 3^{2x} = 3^1 \implies 2x=1\implies x=\frac{1}{2}$,

Assim,  $log_9(3) = \frac{1}{2}$, e

$1+\frac{1}{2}.log_9(3) = 1+\frac{1}{2}.\frac{1}{2} = 1+\frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}$

Logo, $log_9(3.\sqrt{27}) = \frac{5}{4}$.