Question

(Fazu-MG) log 50 + log 40 + log 20 + log 2,5 é igual a:

Answer 1

log 50 + log 40 + log 20 + log 2,5

log 50 = log(2.5^2) ==> log 2 + 2log5

log 40 = log(2^3.5) ==> 3log 2 + log5

log 20 =  log(2^2.5) ==> 2log 2 + log5

log 2,5 =  log(25/10) ==> log(2.5^2) - log10 ==> log 2 + 2log5 - log10
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 log 2 + 2log5 +  3log 2 + log5 +  2log 2 + log5 + log 2 + 2log5 - log10

 1.log 2 +  3log 2 +  2log 2 + log 2 + 2log5 + log5 + 2log5 - log10

 7log 2 + 5log5 - 1

Answer 2

Duas formas de resolver:
$log \ 50 + log \ 40 + log \ 20 + log \ 2,5=log \ (50\cdot40\cdot20\cdot2,5)\\ \\=log \ 100000=log \ 10^5=5\cdot log \ 10=5\cdot 1=5$
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$log \ 50 + log \ 40 + log \ 20 + log \ 2,5\\ \\ =log \ (5\cdot 10) + log \ (4\cdot 10) + log \ (2\cdot 10) + log \ \big( \frac{25}{10}\big)\\ \\= log \ 5+ log \ 10 + log \ 4 +log \ 10 + log \ 2 + log \ 10 + (log \ 25- log \ 10)\\ \\= log \ 5+ log \ 10 + log \ 4 +log \ 10 + log \ 2 + log \ 10 + log \ 25- log \ 10\\ \\= log \ 10 + log \ 10 + log \ 10 - log \ 10+ log \ 4 + log \ 2 + log \ 5+ log \ 25\\ \\=2\ log \ 10 + log \ 2^2 + log \ 2 + log \ 5+ log \ 5^2$
$\\=2\cdot1 + 2\cdot log \ 2 + log \ 2 + log \ 5+ 2\cdot log \ 5\\ \\=2 + 3\cdot log \ 2 + 3\cdot log \ 5\\ \\=2 + 3\cdot log \ 2 + 3\cdot log \ 5\\ \\=2 + 3\cdot log \ 2 + 3\cdot log \ \big( \frac{10}{2} \big)\\ \\=2 + 3\cdot log \ 2 + 3\cdot \big(log \ 10-log \ 2 \big)\\ \\=2 + 3\ log \ 2 + 3\ log \ 10- 3\ log \ 2\\ \\=2 + 3\ log \ 2 + 3\cdot 1- 3\ log \ 2\\ \\=2 + 3\ log \ 2 + 3- 3\ log \ 2\\ \\=2 + 3+ 3\ log \ 2 - 3\ log \ 2\\ \\=5$