1)log12 8 + log12 18 1)log12 48 + log12 3 2)16^log8 27 2)64^log8 7 3)log1/6 36корнь6 3)log1/5 25корень5 4)5^log5 3+1 4)3^1-log3 3 5)log6 (2x+4)=0 5)log7 (3x-2)=3 6)log2^2 x-3log2 x-4=0 6)log3^2 x-8log3 x+12=0

Question

1)log12 8 + log12 18 1)log12 48 + log12 3 2)16^log8 27 2)64^log8 7 3)log1/6 36корнь6 3)log1/5 25корень5 4)5^log5 3+1 4)3^1-log3 3 5)log6 (2x+4)=0 5)log7 (3x-2)=3 6)log2^2 x-3log2 x-4=0 6)log3^2 x-8log3 x+12=0

Yura_Kid 04.02.2026 07:44

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Лебедев Дмитрий Сергеевич · Accepted Answer

Ниже представлены решения для обоих вариантов предложенных заданий. ️ Шаг 1: Вычисление логарифмических выражений (Задания 1–3) Для решения используем свойства логарифмов: $\log_{a} b + \log_{a} c = \log_{a} (b c) log base a of b plus log base a of c equals log base a of b c$ , $a^{\log_{a} b} = b a raised to the log base a of b power equals b$ и $\log_{a^{k}} b^{m} = \frac{m}{k} \log_{a} b log base a to the k-th power of b to the m-th power equals m over k end-fraction log base a of b$ . Вариант 1:

$\log_{12} 8 + \log_{12} 18 = \log_{12} (8 \cdot 18) = \log_{12} 144 = \log_{12} 12^{2} = 2 log base 12 of 8 plus log base 12 of 18 equals log base 12 of open paren 8 center dot 18 close paren equals log base 12 of 144 equals log base 12 of 12 squared equals 2$ $16^{\log_{8} 27} = (2^{4})^{\log_{2^{3}} 3^{3}} = 2^{4 \cdot \frac{3}{3} \log_{2} 3} = 2^{4 \log_{2} 3} = 2^{\log_{2} 3^{4}} = 3^{4} = 81 16 raised to the log base 8 of 27 power equals open paren 2 to the fourth power close paren raised to the exponent log base 2 cubed of 3 cubed end-exponent equals 2 raised to the 4 center dot three-thirds log base 2 of 3 power equals 2 raised to the 4 log base 2 of 3 power equals 2 raised to the exponent log base 2 of 3 to the fourth power end-exponent equals 3 to the fourth power equals 81$ $\log_{1 / 6} 36 \sqrt{6} = \log_{6^{-1}} (6^{2} \cdot 6^{0.5}) = \log_{6^{-1}} 6^{2.5} = \frac{2.5}{-1} = -2.5 log base 1 / 6 of 36 the square root of 6 end-root equals log base 6 to the negative 1 power of open paren 6 squared center dot 6 to the 0.5 power close paren equals log base 6 to the negative 1 power of 6 to the 2.5 power equals 2.5 over negative 1 end-fraction equals negative 2.5$

Вариант 2:

$\log_{12} 48 + \log_{12} 3 = \log_{12} (48 \cdot 3) = \log_{12} 144 = 2 log base 12 of 48 plus log base 12 of 3 equals log base 12 of open paren 48 center dot 3 close paren equals log base 12 of 144 equals 2$ $64^{\log_{8} 7} = (8^{2})^{\log_{8} 7} = 8^{2 \log_{8} 7} = 8^{\log_{8} 7^{2}} = 7^{2} = 49 64 raised to the log base 8 of 7 power equals open paren 8 squared close paren raised to the log base 8 of 7 power equals 8 raised to the 2 log base 8 of 7 power equals 8 raised to the exponent log base 8 of 7 squared end-exponent equals 7 squared equals 49$ $\log_{1 / 5} 25 \sqrt{5} = \log_{5^{-1}} (5^{2} \cdot 5^{0.5}) = \log_{5^{-1}} 5^{2.5} = \frac{2.5}{-1} = -2.5 log base 1 / 5 of 25 the square root of 5 end-root equals log base 5 to the negative 1 power of open paren 5 squared center dot 5 to the 0.5 power close paren equals log base 5 to the negative 1 power of 5 to the 2.5 power equals 2.5 over negative 1 end-fraction equals negative 2.5$

️ Шаг 2: Преобразование степенных выражений (Задание 4) Используем свойства степеней $a^{n + m} = a^{n} \cdot a^{m} a raised to the n plus m power equals a to the n-th power center dot a to the m-th power$ и $a^{n - m} = \frac{a^{n}}{a^{m}} a raised to the n minus m power equals the fraction with numerator a to the n-th power and denominator a to the m-th power end-fraction$ . Вариант 1: 4) $5^{\log_{5} 3 + 1} = 5^{\log_{5} 3} \cdot 5^{1} = 3 \cdot 5 = 15 5 raised to the log base 5 of 3 plus 1 power equals 5 raised to the log base 5 of 3 power center dot 5 to the first power equals 3 center dot 5 equals 15$ Вариант 2: 4) $3^{1 - \log_{3} 3} = 3^{1} ∶ 3^{\log_{3} 3} = 3 ∶ 3 = 1 3 raised to the 1 minus log base 3 of 3 power equals 3 to the first power colon 3 raised to the log base 3 of 3 power equals 3 colon 3 equals 1$ (так как $1 - 1 = 0 1 minus 1 equals 0$ , то $3^{0} = 1 3 to the 0 power equals 1$ ) ️ Шаг 3: Решение логарифмических уравнений (Задания 5–6) Для уравнений вида $\log_{a} f (x) = b log base a of f of x equals b$ переходим к $f (x) = a^{b} f of x equals a to the b-th power$ . Квадратные уравнения решаем через замену переменной. Вариант 1: 5) $\log_{6} (2 x + 4) = 0 ⟹ 2 x + 4 = 6^{0} ⟹ 2 x + 4 = 1 ⟹ 2 x = -3 ⟹ x = -1.5 log base 6 of open paren 2 x plus 4 close paren equals 0 ⟹ 2 x plus 4 equals 6 to the 0 power ⟹ 2 x plus 4 equals 1 ⟹ 2 x equals negative 3 ⟹ bold x equals negative 1.5$ 6) $\log_{2}^{2} x - 3 \log_{2} x - 4 = 0 log base 2 end-base squared of x minus 3 log base 2 of x minus 4 equals 0$ . Пусть $t = \log_{2} x t equals log base 2 of x$ : $t^{2} - 3 t - 4 = 0 ⟹ (t - 4) (t + 1) = 0 t squared minus 3 t minus 4 equals 0 ⟹ open paren t minus 4 close paren open paren t plus 1 close paren equals 0$ $t_{1} = 4 ⟹ \log_{2} x = 4 ⟹ x = 16 t sub 1 equals 4 ⟹ log base 2 of x equals 4 ⟹ bold x equals 16$ $t_{2} = -1 ⟹ \log_{2} x = -1 ⟹ x = 0.5 t sub 2 equals negative 1 ⟹ log base 2 of x equals negative 1 ⟹ bold x equals 0.5$ Вариант 2: 5) $\log_{7} (3 x - 2) = 3 ⟹ 3 x - 2 = 7^{3} ⟹ 3 x - 2 = 343 ⟹ 3 x = 345 ⟹ x = 115 log base 7 of open paren 3 x minus 2 close paren equals 3 ⟹ 3 x minus 2 equals 7 cubed ⟹ 3 x minus 2 equals 343 ⟹ 3 x equals 345 ⟹ bold x equals 115$ 6) $\log_{3}^{2} x - 8 \log_{3} x + 12 = 0 log base 3 end-base squared of x minus 8 log base 3 of x plus 12 equals 0$ . Пусть $t = \log_{3} x t equals log base 3 of x$ : $t^{2} - 8 t + 12 = 0 ⟹ (t - 6) (t - 2) = 0 t squared minus 8 t plus 12 equals 0 ⟹ open paren t minus 6 close paren open paren t minus 2 close paren equals 0$ $t_{1} = 6 ⟹ \log_{3} x = 6 ⟹ x = 729 t sub 1 equals 6 ⟹ log base 3 of x equals 6 ⟹ bold x equals 729$ $t_{2} = 2 ⟹ \log_{3} x = 2 ⟹ x = 9 t sub 2 equals 2 ⟹ log base 3 of x equals 2 ⟹ bold x equals 9$ Ответ: Вариант 1: 1) 2; 2) 81; 3) -2.5; 4) 15; 5) -1.5; 6) 0.5; 16. Вариант 2: 1) 2; 2) 49; 3) -2.5; 4) 1; 5) 115; 6) 9; 729. Нужно ли вам графическое решение для уравнений из шестого пункта или проверка ОДЗ (области допустимых значений) для других примеров?

1)log12 8 + log12 18 1)log12 48 + log12 3 2)16^log8 27 2)64^log8 7 3)log1/6 36корнь6 3)log1/5 25корень5 4)5^log5 3+1 4)3^1-log3 3 5)log6 (2x+4)=0 5)log7 (3x-2)=3 6)log2^2 x-3log2 x-4=0 6)log3^2 x-8log3 x+12=0

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