Question

ТЕМА- ЛОГАРИФМЫ1.log^2(3)x^3-20log(9)x +1=0^- степень(..)-по основанию2. решить неравенствоa)log(6)(x^2+10x+24)<=1+log(6) (x+6)b) log^2(0,5)x-log(0,5)x^2>3подробное решение пожалуйста!

Answer 1

log(9)x=log(3)x/log(3)9=log(3)x/2

9log^2(3)x-10log(3)x+1=0

log(3)x=t

9t^2-10t+1=0

t=(5+-sqrt(25-9))/3=(5+-4)/3

t1=3

t2=1/3

log(3)x=3  x=3^3=27

log(3)x=1/3  x=-1

2. log(6)6+log(6)(x+6)=log(6)(6x+36)

x^2+10x+24<=6x+36

x^2+4x-12<=0

[-6;2]

x>-6 +

x^2+10x+24>0  x>-4  x<-6

 

ответ ]-4;2]

log^2(0,5)x-log(0,5)x^2>3

t^2-2t-3>0

t<-1

t>3

log(0,5)x>3   x<1/8

log(0,5)<-1   x>2

x>0

 ]0;1/8[ U]2;~[

 

Answer 2

Logarithmic Equations:

To solve the given logarithmic equation log^2(3)x^3 - 20log(9)x + 1 = 0, we can use the properties of logarithms and algebraic techniques. Let's break it down step by step:

1. Combine the logarithms using the properties of logarithms: - log^2(3)x^3 - log(9)x^20 + 1 = 0

2. Apply the power rule of logarithms to simplify the equation: - log(3)(x^6) - log(9)x^20 + 1 = 0

3. Use the quotient rule of logarithms to combine the logarithms: - log(3)((x^6)/(9x^20)) + 1 = 0

4. Simplify the expression inside the logarithm: - log(3)((x^(-14))/9) + 1 = 0

5. Apply the logarithmic identity log(a)b = c is equivalent to b = a^c: - (x^(-14))/9 = 3^(-1)

6. Solve for x by isolating the variable: - Multiply both sides of the equation by 9: - (x^(-14)) = 3^(-1) * 9 - (x^(-14)) = 3^(-1) * 3^2 - (x^(-14)) = 3^1 - (x^(-14)) = 3 - Take the reciprocal of both sides to get rid of the negative exponent: - x^14 = 1/3 - Take the 14th root of both sides to solve for x: - x = (1/3)^(1/14)

Therefore, the solution to the equation log^2(3)x^3 - 20log(9)x + 1 = 0 is x = (1/3)^(1/14).

Logarithmic Inequalities:

Now let's solve the given logarithmic inequalities:

a) log(6)(x^2 + 10x + 24) <= 1 + log(6)(x + 6)

To solve this inequality, we can use the properties of logarithms and algebraic techniques. Here's the step-by-step solution:

1. Combine the logarithms using the properties of logarithms: - log(6)(x^2 + 10x + 24) - log(6)(x + 6) <= 1

2. Apply the quotient rule of logarithms to simplify the inequality: - log(6)((x^2 + 10x + 24)/(x + 6)) <= 1

3. Rewrite the inequality using the logarithmic identity log(a)b <= c is equivalent to b <= a^c: - (x^2 + 10x + 24)/(x + 6) <= 6^1

4. Simplify the expression inside the logarithm: - (x^2 + 10x + 24)/(x + 6) <= 6

5. Solve the inequality by considering two cases: - Case 1: x + 6 > 0 (no need to flip the inequality) - x^2 + 10x + 24 <= 6(x + 6) - x^2 + 10x + 24 <= 6x + 36 - x^2 + 4x - 12 <= 0 - (x + 6)(x - 2) <= 0 - The solution for this case is -6 <= x <= 2.

- Case 2: x + 6 < 0 (flip the inequality) - (x^2 + 10x + 24)/(x + 6) >= 6 - x^2 + 10x + 24 >= 6(x + 6) - x^2 + 10x + 24 >= 6x + 36 - x^2 + 4x - 12 >= 0 - (x + 6)(x - 2) >= 0 - The solution for this case is x <= -6 or x >= 2.

- Combining the solutions from both cases, the solution to the inequality is x <= -6 or -6 <= x <= 2.

b) log^2(0.5)x - log(0.5)x^2 > 3

To solve this inequality, we can use the properties of logarithms and algebraic techniques. Here's the step-by-step solution:

1. Combine the logarithms using the properties of logarithms: - log^2(0.5)x - log(0.5)x^2 > 3

2. Apply the logarithmic identity log(a)b - log(a)c = log(a)(b/c): - log(0.5)(x^2/x^2) > 3

3. Simplify the expression inside the logarithm: - log(0.5)(1/x^2) > 3

4. Apply the logarithmic identity log(a)b > c is equivalent to b > a^c: - 1/x^2 > 0.5^3

5. Simplify the right side of the inequality: - 1/x^2 > 0.125

6. Take the reciprocal of both sides to get rid of the fraction: - x^2 < 1/0.125

7. Simplify the right side of the inequality: - x^2 < 8

8. Take the square root of both sides to solve for x: - -√8 < x < √8

Therefore, the solution to the inequality log^2(0.5)x - log(0.5)x^2 > 3 is -√8 < x < √8.

Please note that these solutions are based on the given equations and inequalities. Make sure to double-check the calculations and consider any restrictions on the domain of the logarithmic functions.