A expressão log a10(7m) * log a12(a10) * log a13(a12) * log7(a13) com m inteiro, é igual a: a) log 7 b) log m7 c) a^11 d) 7^m e) m

Podemos utilizar a propriedade de mudança de base dos logaritmos para resolver essa questão. log a10(7m) * log a12(a10) * log a13(a12) * log7(a13) = (log 7m / log 10a) * (log a10 / log 12a) * (log a12 / log 13a) * (log 13a / log 7) Podemos simplificar algumas expressões: log a10 = 1 log 10a = log 10 + log a = 1 + log a log a12 = log a * log 12 log 12a = log 12 + log a log 13a = log 13 + log a Substituindo na expressão original: (log 7m / log 10a) * (1 / (log 12 / log a)) * (log a * log 12 / (log 13 / log a)) * ((log 13 + log a) / log 7) (log 7m / (1 + log a)) * (log a * log 12 / log 13) * ((log 13 + log a) / log 7) (log 7m / (1 + log a)) * (log a * log 12 / log 13) * ((log 13 / log 7) + (log a / log 7)) (log 7m / (1 + log a)) * (log a * log 12 / log 13) * (log 7a / log 7) (log 7m / (1 + log a)) * (log a * log 12 / log 13) * log a (log 7m * log a * log a * log 12) / (log 13 * (1 + log a) * log 7) (log 7m * log a^2 * log 12) / (log 13 * (1 + log a) * log 7) (log 7m * 2 * log a * log 12) / (log 13 * (1 + log a) * log 7) (log 7m * log a * log 12) / (log 13 * (1 + log a) * log 7/2) (log 7m * log a * log 12) / (log 13 * (1 + log a) * log 49) (log 7m * log a * log 12) / (log 13 * log 7^2 * (1 + log a)) (log 7m * log a * log 12) / (2 * log 7 * log a * (1 + log a)) (log 7m * log 6) / (2 * log 7 * (1 + log a)) (log 7m * log 2 * log 3) / (2 * log 7 * (1 + log a)) (log 7m * log 2 * log 3) / (2 * log 7 + 2 * log 7 * log a) (log 7m * log 2 * log 3) / (2 * log 7 * (1 + log a)) A resposta correta é a letra E) m.