SPMB 2004 Regional II (Logaritma)

$\frac{\left (^{5}\textrm{log}\: 10 \right )^{2}-\left ( ^{5}\textrm{log}\: 2 \right )^{2}}{^{5}\textrm{log}\: \sqrt{20}}$ = ....

(A) $\frac{1}{2}$
(B) 1
   (C) 2
   (D) 4
   (E) 5

Pembahasan:
Gunakan  $a^{2}-b^{2}=\left ( a+b \right )\left ( a-b \right )$ 

   $\frac{\left (^{5}\textrm{log}\: 10 \right )^{2}-\left ( ^{5}\textrm{log}\: 2 \right )^{2}}{^{5}\textrm{log}\: \sqrt{20}}$ 
$=\frac{\left (^{5}\textrm{log}\: 10+\: ^{5}\textrm{log}\: 2 \right )\left ( ^{5}\textrm{log}\: 10-\: ^{5}\textrm{log}\: 2 \right )}{^{5}\textrm{log}\: 20^{\frac{1}{2}}}$
$=\frac{\left (^{5}\textrm{log}\: 10\: \cdot \: 2 \right )\left ( ^{5}\textrm{log}\: \frac{10}{2}\right )}{\frac{1}{2}\: ^{5}\textrm{log}\: 20}$
$=\frac{\left (^{{\color{Red} 5}}\textrm{log}\: {\color{Red} 20} \right )\left ( ^{{\color{Blue} 5}}\textrm{log}\: {\color{Blue} 5}\right )}{\frac{1}{2}\: ^{{\color{Red} 5}}\textrm{log}\: {\color{Red} 20}}$
$=\frac{1}{\frac{1}{2}}$
$=2$

Jawaban __________________________________ (C)