log 6​ (2x 2 −7)−log 6​ (x+3)=0

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

The equation is: log6(2x^2 - 7) - log6(x + 3) = 0

Step 1: Use the properties of logarithms to combine the two logs into one. The subtraction of logarithms becomes division inside the log.

log6[(2x^2 - 7) / (x + 3)] = 0

Step 2: The log base 6 of a number is 0 if and only if the number is 1. So, we can set the argument of the log equal to 1.

(2x^2 - 7) / (x + 3) = 1

Step 3: Solve the equation for x. First, multiply both sides by (x + 3) to get rid of the denominator.

2x^2 - 7 = x + 3

Step 4: Rearrange the equation to get a quadratic equation.

2x^2 - x - 10 = 0

Step 5: Factor the quadratic equation.

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

Step 6: Set each factor equal to zero and solve for x.

2x + 5 = 0 => x = -5/2

x - 2 = 0 => x = 2

Step 7: Check the solutions in the original equation. x = -5/2 is not a valid solution because it would make the argument of the second log negative. So, the only solution is x = 2.