Question

Log3(2x-1)+log3(x+3)=2 log0,2x(меньше или равно)-3

Answer 1

#1
$log_{_{3}}(2x-1) + log_{_{3}}(x+3) = 2 \\ \\ log_{_{3}}((2x-1)(x+3)) = 2 \\ \\ log_{_{3}}(2x^{2}+5x-3) = 2 \\ \\ 2x^{2}+5x-3=9 \\ \\ 2x^{2}+5x-12=0 \\ D = 25 + 4*2*12 = 25 +96 = 121 \\ \\ x_{1} = \dfrac{-5+11}{4} = \dfrac{3}{2} \ ; \ x_{2} = -4$
Проверка: 
$x = \dfrac{3}{2}: \ \ log_{_{3}}(2 *1.5-1) + log_{_{3}}(1.5+3) = 2 \ \rightarrow \ 2 = 2 \\ \\ x = -4: \ \ log_{_{3}}(2*(-4)-1) + log_{_{3}}(-4+3) = 2 \ \rightarrow \ Reshenia\ net$
Ответ: x = 1.5

#2
$log_{_{0,2}}x\leq -3 \\ log_{_{5^{(-1)}}}x\leq -3 \\ -log_{_{5}}x\leq -3 \ \ /*(-1) \\ x \geq 5^{3} \\ x \geq 125 \\ x\in [125;+\infty)$
Ответ: x∈[125;+∞)

Answer 2

Let's solve the given logarithmic equation step by step.

The given equation is: log₃(2x - 1) + log₃(x + 3) = 2

Step 1: Combine the logarithms using the product rule of logarithms: log₃((2x - 1) * (x + 3)) = 2

Step 2: Convert the equation into exponential form: 3^2 = (2x - 1) * (x + 3)

Step 3: Simplify and solve the quadratic equation: 9 = 2x^2 + 5x - 3

Step 4: Move all terms to one side of the equation: 2x^2 + 5x - 12 = 0

Step 5: Factor the quadratic equation (or use the quadratic formula): (2x - 3)(x + 4) = 0

Step 6: Set each factor to zero and solve for x:

1. 2x - 3 = 0 2x = 3 x = 3/2

2. x + 4 = 0 x = -4

So, the solutions to the equation are x = 3/2 and x = -4.

Now, let's solve the inequality: log₀.₂x ≤ -3

Step 1: Convert the inequality into exponential form: ₀.₂x ≤ ₂₋³ (Note: ₀.₂ represents the base of the logarithm, which is 1/5)

Step 2: Rewrite ₂₋³ as 1/(2^3) = 1/8: ₀.₂x ≤ ₁/₈

Step 3: Convert the base of the logarithm to a common base, such as 10: log₁/₅x ≤ log₁/₁₀(₁/₈)

Step 4: Evaluate the logarithm on the right side: log₁/₅x ≤ log₁/₁₀(₁/₈) = -log₁₀(₈) = -(-3) = 3

So, the solution to the inequality is x ≤ 3.

Please note that the solution x = -4 should be discarded as it doesn't satisfy the original logarithmic equation (you cannot take the logarithm of a non-positive value), leaving the valid solution x = 3/2.