Question

Log5(x-3)+2=log5(x-2)+log3( 5)

Answer 1

Ответ:

$x=\dfrac{2\cdot5^{\log_35}+75}{25+5^{\log_35}}$

Пошаговое объяснение:

1 Запишем уравнение

$></p> <p><strong>2</strong> Запишем ОДЗ: аргумент логарифма строго больше 0</p> <p><img src=$

3 Решим эту систему неравенств

$\displaystyle \left \{ {{3-x>0} \atop {x-2>0}} \right.\Leftrightarrow \left \{ {{3>x} \atop {x>2}} \right.\Leftrightarrow x\in(2;3)$

4 Запишем 2 как $\log_525$ и $\log_35$ как $\log_55^{\log_35}$

$\log_5(3-x)+\log_525=\log_5(x-2)+\log_55^{\log_35}$

5 Воспользуемся формулой $\log_ab+\log_ac=\log_a(bc)$

$\log_5\bigg((3-x)\cdot25\bigg)=\log_5\bigg((x-2)\cdot5^{\log_35}\bigg)$

6 Уберем логарифмы (все ограничения записаны в ОДЗ)

$(3-x)\cdot25=(x-2)\cdot5^{\log_35}$

7 Перед нами линейное уравнение

$75-25x=5^{\log_35}x-2\cdot5^{\log_35}$

8 Все что с х налево, все что без направо

$-25x-5^{\log_35}x=-2\cdot5^{\log_35}-75$

9 Умножим все на -1 и вынесем х

$\big(25+5^{\log_35}\big)x=2\cdot5^{\log_35}+75$

10 Выразим х

$x=\dfrac{2\cdot5^{\log_35}+75}{25+5^{\log_35}}$

11 Оценим значение, чтобы понять, принадлежит ли оно промежутку

$\dfrac{2\cdot5^{\log_35}+75}{25+5^{\log_35}}\approx\dfrac{2*10.6+75}{25+10.6}\approx\dfrac{96.13}{35.57}\approx2.703$

Принадлежит промежутку.

Тогда это ответ!

Answer 2

To solve the equation Log5(x-3) + 2 = Log5(x-2) + Log3(5), you can use logarithmic properties to simplify it and isolate the variable x. Here are the steps to solve it:

Step 1: Apply the properties of logarithms.

Log5(x-3) + 2 = Log5(x-2) + Log3(5)

Step 2: Use the fact that Log5(a) + Log5(b) = Log5(a * b) and subtract Log3(5) from both sides to combine the logarithms on the right side:

Log5(x-3) + 2 - Log3(5) = Log5(x-2)

Step 3: Apply the property that Log5(a) - Log5(b) = Log5(a/b) to the left side of the equation:

Log5[(x-3)/5] + 2 - Log3(5) = Log5(x-2)

Step 4: Now, you can eliminate the 2 and the Log3(5) from the left side by subtracting them:

Log5[(x-3)/5] - Log3(5) = Log5(x-2)

Step 5: Apply the property that Log5(a) - Log5(b) = Log5(a/b) again to combine the logarithms on the left side:

Log5[((x-3)/5) / 3] = Log5(x-2)

Step 6: Now that the bases are the same, you can equate the arguments inside the logarithms:

((x-3)/5) / 3 = x-2

Step 7: Solve for x:

First, simplify the left side:

((x-3)/5) / 3 = ((x-3)/5) * (1/3) = (x-3)/15

Now, set this equal to x-2:

(x-3)/15 = x-2

Step 8: Solve for x:

Multiply both sides of the equation by 15 to eliminate the fraction:

x-3 = 15(x-2)

x-3 = 15x - 30

Step 9: Move the x terms to one side of the equation and constants to the other side:

x - 15x = -30 + 3

-14x = -27

Step 10: Divide by -14 to isolate x:

x = (-27) / (-14)

x = 27/14

So, the solution to the equation is:

x = 27/14