Question

Log4x=2; Log2(1-3x)=3;Lg(x-3)=Lg(2x-4); Log5x=-2;Log1/3(2-x)=-2;Log2(2x+1)=Log2 3​

Answer 1

Відповідь:

Покрокове пояснення:

ОДЗ: многочлен под логорифмом должен быть больше 0

Log4 x=2;->х=4² -> х =16 и ОДЗ: х>0. Ответ: х=16

Log2 (1-3x)=3;-> 1-3х=2³ -> -3х=8-1, х=-7/3=-2⅓ и ОДЗ: 1-3х>0, х<1/3. Ответ: х= -2⅓

Lg(x-3)=Lg(2x-4); -> х-3=2х-4, х=1 и ОДЗ: х>3 Ответ: корней нет

Log5 x=-2;-> х=1/5²=1/25=0,04 и ОДЗ: х>0. Ответ: х=0.04

Log1/3 (2-x)=-2; -> 2-х=3² х=‐7 и ОДЗ: х<2. Ответ: х=-7

Log2 (2x+1)=Log2 3​ -> 2х+1=3 ->х=1 и ОДЗ: 2х+1>0, х>-1/2. Ответ: х=1

Answer 2

To solve the given equations, let's take them one by one:

1. Log4x = 2: To solve for x, we need to rewrite the equation in exponential form. The base of the logarithm is 4, and the result is 2. So, we have: 4^2 = x 16 = x

Therefore, the solution to the equation is x = 16.

2. Log2(1-3x) = 3: Similar to the previous equation, we need to rewrite it in exponential form. The base of the logarithm is 2, and the result is 3. So, we have: 2^3 = 1 - 3x 8 = 1 - 3x

Now, let's solve for x: 3x = 1 - 8 3x = -7 x = -7/3

Therefore, the solution to the equation is x = -7/3.

3. Lg(x-3) = Lg(2x-4): Here, "Lg" is most likely a typo and should be "log" or "log10". Assuming it's a base 10 logarithm, we can rewrite the equation as: log10(x - 3) = log10(2x - 4)

Since the logarithms have the same base, we can equate the arguments: x - 3 = 2x - 4

Now, solve for x: x - 2x = -4 + 3 -x = -1 x = 1

Therefore, the solution to the equation is x = 1.

4. Log5x = -2: Rewrite the equation in exponential form with the base 5 and the result -2: 5^(-2) = x 1/25 = x

So, the solution to the equation is x = 1/25.

5. Log1/3(2-x) = -2: Rewrite the equation in exponential form with the base 1/3 and the result -2: (1/3)^(-2) = 2 - x

Simplify the left side: 3^2 = 2 - x 9 = 2 - x

Now, solve for x: x = 2 - 9 x = -7

Therefore, the solution to the equation is x = -7.

6. Log2(2x + 1) = Log2 + 3: Rewrite the equation in exponential form with the base 2 on both sides: 2^(Log2(2x + 1)) = 2^(Log2 + 3)

Simplify the left side using the logarithmic identity: 2x + 1 = 2 * 2^3 2x + 1 = 16

Now, solve for x: 2x = 16 - 1 2x = 15 x = 15/2

Therefore, the solution to the equation is x = 15/2.

To summarize: - Log4x = 2: x = 16 - Log2(1-3x) = 3: x = -7/3 - Lg(x-3) = Lg(2x-4): x = 1 - Log5x = -2: x = 1/25 - Log1/3(2-x) = -2: x = -7 - Log2(2x+1) = Log2 + 3: x = 15/2