Question

Log2/3x-2log3(x-8)=0​

Answer 1

Ответ:

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Answer 2

To solve the equation log2/3(x) - 2log3(x - 8) = 0, you can follow these steps:

Step 1: Use logarithmic properties to simplify the equation.

log2/3(x) - log3((x - 8)^2) = 0

Step 2: Combine the logarithms using the properties of logarithms.

log2/3(x / ((x - 8)^2)) = 0

Step 3: Remove the logarithm by rewriting the equation in exponential form.

2/3(x / ((x - 8)^2)) = 1

Step 4: Multiply both sides of the equation by 3/2 to isolate the expression inside the parentheses.

x / ((x - 8)^2) = 3/2

Step 5: Cross-multiply to get rid of the fraction.

2x = 3 * ((x - 8)^2)

Step 6: Expand and simplify the right side of the equation.

2x = 3 * (x^2 - 16x + 64)

Step 7: Distribute 3 on the right side.

2x = 3x^2 - 48x + 192

Step 8: Move all the terms to one side to set the equation equal to zero.

3x^2 - 48x + 192 - 2x = 0

Step 9: Combine like terms.

3x^2 - 50x + 192 = 0

Step 10: To further simplify the equation, you can divide the entire equation by 3 to make the coefficients smaller.

x^2 - (50/3)x + 64 = 0

Step 11: Now, you can attempt to factor the quadratic equation. However, it may not factor easily. You can use the quadratic formula to find the solutions.

The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -(50/3), and c = 64.

x = (-(50/3) ± √((-(50/3))^2 - 4 * 1 * 64)) / (2 * 1)

x = (-(50/3) ± √(2500/9 - 256)) / 2

Now, calculate the square root and simplify further:

x = (-(50/3) ± √((2500/9 - 2304/9))) / 2

x = (-(50/3) ± √(196/9)) / 2

x = (-(50/3) ± 14/3) / 2

Now, you have two possible solutions:

1. x = (-(50/3) + 14/3) / 2
2. x = (-(50/3) - 14/3) / 2

Simplify these to find the values of x:

1. x = (-(50/3) + 14/3) / 2 = (-36/3 + 14/3) / 2 = (-22/3) / 2 = -11/3

2. x = (-(50/3) - 14/3) / 2 = (-64/3) / 2 = -32/3

So, the solutions to the equation are:

x = -11/3 and x = -32/3