Question

Sabendo ㏒ 2 = x e ㏒ 3 = y, calcule

㏒$_{2}$ 500

Answer 1

$log_2 \ 500=\frac{log \ 500}{log \ 2}=\frac{log \ (5 \cdot 100)}{log \ 2}=\frac{log \ 5 \ + \ log \ 100}{log \ 2}=\frac{log \ 5 \ + \ log \ 10^2}{log \ 2}\\ \\=\frac{log \ 5 \ + \ log \ 10^2}{log \ 2}=\frac{log \ (\frac{10}{2}) \ + \ 2\ \cdot\ log \ 10}{log \ 2}=\frac{(log \ 10\ - \ log\ 2) \ + \ 2\ \cdot\ log \ 10}{log \ 2}\\ \\=\frac{(1\ - \ x) \ + \ 2\ \cdot\ 1}{x}=\frac{1\ - \ x \ + \ 2}{x}=\frac{3\ - \ x}{x}$
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$log_2 \ 500=\frac{log \ 500}{log \ 2}=\frac{log \ (2^2\cdot5^3)}{log \ 2}=\frac{log \ 2^2 \ + \ log \ 5^3}{log \ 2}=\frac{2log \ 2 \ + \ 3log \ 5}{log \ 2}\\ \\=\frac{2log \ 2 \ + \ 3log \ (\frac{10}{2})}{log \ 2}=\frac{2x\ + \ 3 \cdot(log \ 10 - \ log \ 2)}{x}=\frac{2x\ + \ 3\cdot(1 - \ x)}{x}\\ \\=\frac{2x\ + \ 3 - 3x}{x}=\frac{3 - x}{x}$