Question

Log3(2x-1)=2-log 3(x-4)

Answer 1

во вкладке..................

Answer 2

To solve the equation log3(2x - 1) = 2 - log3(x - 4), we'll use the properties of logarithms to simplify and solve for x.

Step 1: Combine the logarithms on the right side of the equation using the logarithm property log(a) - log(b) = log(a/b):

log3(2x - 1) = 2 - log3(x - 4) log3(2x - 1) + log3(x - 4) = 2

Step 2: Apply the logarithm property log(a) + log(b) = log(a * b):

log3((2x - 1) * (x - 4)) = 2

Step 3: Remove the logarithm by exponentiating both sides with base 3:

3^2 = (2x - 1) * (x - 4)

Step 4: Simplify and solve for x:

9 = 2x^2 - 9x + 4

Step 5: Rearrange the equation in standard quadratic form:

2x^2 - 9x + 4 - 9 = 0

2x^2 - 9x - 5 = 0

Step 6: Solve the quadratic equation. You can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In our equation, a = 2, b = -9, and c = -5:

x = (9 ± √((-9)^2 - 4 * 2 * (-5))) / 2 * 2

x = (9 ± √(81 + 40)) / 4

x = (9 ± √121) / 4

Now, there are two solutions for x:

1. x = (9 + 11) / 4 x = 20 / 4 x = 5

2. x = (9 - 11) / 4 x = -2 / 4 x = -0.5

So, the solutions to the equation are x = 5 and x = -0.5. However, we need to check if these solutions are valid in the original equation since logarithms are only defined for positive values. Let's check:

1. For x = 5:

log3(2(5) - 1) = log3(9) = 2 2 - log3(5 - 4) = 2 - log3(1) = 2 - 0 = 2

Both sides of the equation are equal when x = 5.

2. For x = -0.5:

log3(2(-0.5) - 1) = log3(-2) (Logarithm of a negative number is undefined in real numbers) 2 - log3(-0.5 - 4) = 2 - log3(-4.5) (Logarithm of a negative number is undefined in real numbers)

Since log3(-2) and log3(-4.5) are undefined in the real number system, x = -0.5 is not a valid solution.

Thus, the only valid solution to the equation is x = 5.