Use as propriedades do logaritmo para simplificar a expressão: S = + 1 + 1 valor de 2016 2 + + 5 5 2 log 2016 + Slog 2016 + 10log 2016 = log42 2016...

Vamos começar simplificando a expressão S: S = log2016(2^1 * 5^1) + log2016(2^5 * 5^2) + Slog2016 + 10log2016 S = log2016(10) + log2016(500) + Slog2016 + log2016(10^10) S = log2016(10 * 500 * 10^10) + Slog2016 S = log2016(5 * 10^12) + Slog2016 Agora, substituindo S na equação dada: log2016(42) = log2016(5 * 10^12) + Slog2016 + 10log2016 log2016(42) = log2016(5 * 10^12 * 10^10) + Slog2016 log2016(42) = log2016(5 * 10^22) + Slog2016 log2016(42) = log2016(2^22.5 * 5^22) + Slog2016 log2016(42) = 22.5log2016(2) + 22log2016(5) + Slog2016 log2016(42) = 22.5 * log2016(2) + 22 * log2016(5) + Slog2016 Agora, igualando as expressões de S: log2016(5 * 10^12) + Slog2016 = 22.5 * log2016(2) + 22 * log2016(5) + Slog2016 log2016(5 * 10^12) = 22.5 * log2016(2) + 22 * log2016(5) log2016(5) + log2016(10^12) = 22.5 * log2016(2) + 22 * log2016(5) log2016(5) + 12 = 22.5 * log2016(2) + 22 * log2016(5) log2016(5) - 22 * log2016(5) = 22.5 * log2016(2) - 12 -21 * log2016(5) = 22.5 * log2016(2) - 12 log2016(5^-21) = log2016(2^(45/2)) - log2016(2^2) log2016(1/5^21) = log2016(2^(41/2)) 1/5^21 = 2^(41/2) 1/2^(82) = 5^42 Portanto, a resposta é a alternativa E) 5^42.