Solve for x in the equation ln(16-12x)-2lnx)= ln4

ln [ (16-12x)/x^2 ] = ln 4

(16-12x)/x^2 = 4
4x^2 = 16-12x
x^2 + 3x - 4 = 0
(x+4)(x-1) = 0
x = -4 or x=1

but if x = -4 , 2lnx in the original equation is not defined, so

x = 1

To solve for x in the equation ln(16-12x) - 2ln(x) = ln(4), we can follow these steps:

Step 1: Simplify the equation by using the properties of logarithms. Specifically, use the logarithmic property ln(a) - ln(b) = ln(a/b), and move the ln(4) to the other side of the equation:

ln(16-12x) - 2ln(x) = ln(4)
ln(16-12x)/x^2 = ln(4)
ln((16-12x)/x^2) = ln(4)

Step 2: Since the logarithmic function is one-to-one, we can remove the natural logarithms of both sides to obtain:

(16-12x)/x^2 = 4

Step 3: Multiply both sides by x^2 to eliminate the denominator:

16-12x = 4x^2

Step 4: Rearrange the equation into a quadratic form by moving all terms to one side:

4x^2 + 12x - 16 = 0

Step 5: Divide the equation by a common factor, in this case, 4:

x^2 + 3x - 4 = 0

Step 6: Factor the quadratic equation:

(x + 4)(x - 1) = 0

Step 7: Set each factor equal to zero to solve for x:

x + 4 = 0 or x - 1 = 0

Step 8: Solve for x:

x = -4 or x = 1

Therefore, the solutions to the equation ln(16-12x) - 2ln(x) = ln(4) are x = -4 and x = 1.