Question

sabendo que log3 (7x - 1 ) = 3 e que log2 (y3+3) = 7 pode-se afirmar que logy(x2+9) é igaual a

Answer 1

Olá Matheus.


Checando a condição de existência do logaritmando.


$\mathsf{C.E:~7x-1\ \textgreater \ 0~\Rightarrow~7x\ \textgreater \ 1~\Rightarrow~x=\dfrac{1}{7}}$

Encontrando os valores das incógnitas.

$\mathsf{\ell og_3(7x-1)=3~\Rightarrow~7x-1=3^3~\Rightarrow 7x-1=27}\\\\\mathsf{7x=27+1~\Rightarrow~x=\dfrac{28}{7}~\Rightarrow~\boxed{\mathsf{x=4}}}$

$\mathsf{\underline{\qquad\qquad\qquad\qquad}}$

$\mathsf{\ell og_2(y^3+3) = 7~\Rightarrow~y^3+3=2^7~\Rightarrow y^3+3=128}\\\\\mathsf{y^3=128-3~\Rightarrow \sqrt[3]{\mathsf{y^3}}=\sqrt[3]{\mathsf{125}}~\Rightarrow~y=5}$

$\mathsf{\underline{\qquad\qquad\qquad\qquad}}$

$\mathsf{\ell og_y(x^2+9)~\Rightarrow~\ell og_5(4^2+9)~\Rightarrow~\ell og_5(16+9)~\Rightarrow~\ell og_5(25)}\\\\\mathsf{\ell og_5(5^2)~\Rightarrow~2\cdot \ell og_5(5)~\Rightarrow~\boxed{\mathsf{\ell og_y(x^2+9)=2}}}$

$\mathsf{\underline{\qquad\qquad\qquad\qquad}}$


Dúvidas? comente.