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Консультация № 196981
05.11.2019, 23:33
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Сравнить числа.
Сравнить числа.
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Обсуждение
06.11.2019, 16:52
общий
это ответ
Здравствуйте, svrvsvrv!
Дано : Сравнить 2 математических выражения log(1/2)(10) и log(1/3)(10) по величине.
Решение : Используем популярнейшее свойство логарифмов loga(b) = lg(b)/lg(a) = ln(b)/ln(a) , описанное во всех школьных учебниках по математике для старших классов и в справочниках по математике, например "Справочник по математике" 15,3 МБ стр 44 Ссылка1
Это свойство позволяет решить Вашу задачу в одно действие :
log(1/2)(10) = ln(10)/ln(1/2) = -3,311 < log(1/3)(10) = ln(10)/ln(1/3) = -2,096
Здесь ln(b) - натуральный логарифм числа b (по основанию числа e), а lg(b) - десятичный логарифм числа b (по основанию числа 10).
Функция вычисления натурального логарифма ln(b) имеется в апплете Калькулятор любой операционной системы Windows , если переключить этот Калькулятор из режима Обычный в режим Инженерный.
Другое свойство логарифмов loga(b) = 1 / logb(a) позволяет решить Вашу задачу вообще без вычислений, потому что график логарифмической функции F(x) = lg(x) - это монотонно-возрастающая кривая, где 1/2 > 1/3 , и поэтому lg(1/2) > lg(1/3) .
Значит log(1/2)(10) = 1/lg(1/2) < log(1/3)(10) = 1/lg(1/3)
Ответ : log(1/2)(10) < log(1/3)(10)
Дано : Сравнить 2 математических выражения log(1/2)(10) и log(1/3)(10) по величине.
Решение : Используем популярнейшее свойство логарифмов loga(b) = lg(b)/lg(a) = ln(b)/ln(a) , описанное во всех школьных учебниках по математике для старших классов и в справочниках по математике, например "Справочник по математике" 15,3 МБ стр 44 Ссылка1
Это свойство позволяет решить Вашу задачу в одно действие :
log(1/2)(10) = ln(10)/ln(1/2) = -3,311 < log(1/3)(10) = ln(10)/ln(1/3) = -2,096
Здесь ln(b) - натуральный логарифм числа b (по основанию числа e), а lg(b) - десятичный логарифм числа b (по основанию числа 10).
Функция вычисления натурального логарифма ln(b) имеется в апплете Калькулятор любой операционной системы Windows , если переключить этот Калькулятор из режима Обычный в режим Инженерный.
Другое свойство логарифмов loga(b) = 1 / logb(a) позволяет решить Вашу задачу вообще без вычислений, потому что график логарифмической функции F(x) = lg(x) - это монотонно-возрастающая кривая, где 1/2 > 1/3 , и поэтому lg(1/2) > lg(1/3) .
Значит log(1/2)(10) = 1/lg(1/2) < log(1/3)(10) = 1/lg(1/3)
Ответ : log(1/2)(10) < log(1/3)(10)
5
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