log 𝑥 + log(𝑥 − 1) = log(3𝑥 + 2)

Sure, let's solve the equation step by step.

The equation is: log 𝑥 + log(𝑥 − 1) = log(3𝑥 + 2)

Step 1: Use the property of logarithms that states log(a) + log(b) = log(ab). So, the left side of the equation becomes log[x(x-1)].

So, the equation now is: log[x*(x-1)] = log(3𝑥 + 2)

Step 2: If log(a) = log(b), then a = b. So, we can equate x*(x-1) and 3x+2.

So, the equation now is: x*(x-1) = 3x + 2

Step 3: Simplify the equation.

x^2 - x = 3x + 2

Step 4: Rearrange the equation to one side to set it to zero.

x^2 - 4x - 2 = 0

Step 5: Solve the quadratic equation using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a).

Here, a = 1, b = -4, and c = -2.

So, the solutions for x are: x = [4 ± sqrt((-4)^2 - 41(-2))] / (2*1)

Solving this gives the roots of the equation, which are the solutions for x.