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

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To solve this equation, we can apply the properties of logarithms:

Use the property loga(b) = c is equivalent to a^c = b.

Given log3(2x-1) = 2 - log3(x-4), we can rewrite this as:

3^(2 - log3(x-4)) = 2x - 1

Use the property loga(b) - loga(c) = loga(b/c):

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

9 / (x-4) = 2x - 1

Multiply both sides by (x-4) to get rid of the denominator:

9 = (2x - 1)(x - 4)

Expand and simplify:

9 = 2x^2 - 9x + 4

2x^2 - 9x - 5 = 0

Factor the quadratic equation:

(2x + 1)(x - 5) = 0

Set each factor equal to zero:

2x + 1 = 0 or x - 5 = 0

Solve for x:

2x = -1 or x = 5

x = -1/2 or x = 5

Therefore, the solutions to the equation log3(2x-1) = 2 - log3(x-4) are x = -1/2 or x = 5.