Question

28)O valor de $\mathsf{log_{8}[ \dfrac{ (\sqrt{a}+ \sqrt{b})^{2}}{ \sqrt{ab} }] }$

Sabendo que:
$\mathsf{a+b=30 \sqrt{ab} }$

OBRIGADO

Answer 1

$log _{8} \frac{( \sqrt{a} + \sqrt{b})^{2}}{\sqrt{ab}}$
$log _{8} \frac{a + 2\sqrt{ab} + b}{\sqrt{ab}}$
$log _{8} \frac{a + b + 2\sqrt{ab}}{\sqrt{ab}}$

Como a + b = 30$\sqrt{ab}$

$log _{8} \frac{30\sqrt{ab} + 2\sqrt{ab}}{\sqrt{ab}}$
$log _{8} \frac{32\sqrt{ab}}{\sqrt{ab}}$
$log _{8} 32$

$log _{8} 32 = x$
8ˣ = 32
2³ˣ = 2⁵
3x = 5
x = 5/3

=)

Answer 2

$log _{8} \frac{( \sqrt{a} + \sqrt{b})^{2}}{\sqrt{ab}}$

$log _{8} \frac{a + 2\sqrt{ab} + b}{\sqrt{ab}}$

$log _{8} \frac{a + b + 2\sqrt{ab}}{\sqrt{ab}}$

note que $\mathsf{ a + b = 30\sqrt{ab}}$ substituindo temos:

$log _{8} \dfrac{30\sqrt{ab} + 2\sqrt{ab}}{\sqrt{ab}}$

$log _{8} \dfrac{32\sqrt{ab}}{\sqrt{ab}}$$log _{8} 32=log_{{2}^{3}} {2}^{5}=\boxed{\boxed{\dfrac{5}{3}}}$