log4 (x-2) - log4 (x+1) = 1
log4(x-2)(x+1) = 1
(x-2)(x+1) = 4
x^2 - x - 6 = 0
(x-3)(x+2)
x = 3 or -2
however, x = -2 does not fit the original equation, since log of negative numbers are not defined.
So, the only solution is x=3
To solve the equation log4 (x-2) - log4 (x+1) = 1, we will use logarithmic properties and algebraic manipulation.
Step 1: Combine the logarithms using the quotient rule of logarithms.
log4((x-2)/(x+1)) = 1
Step 2: Rewrite the equation using exponentiation.
4^1 = (x-2)/(x+1)
Step 3: Simplify the left side of the equation.
4 = (x-2)/(x+1)
Step 4: Multiply both sides of the equation by (x+1) to eliminate the fraction.
4(x+1) = x-2
Step 5: Distribute on the left side of the equation.
4x + 4 = x - 2
Step 6: Move all the x terms to one side and the constant terms to the other side.
4x - x = -2 - 4
3x = -6
Step 7: Divide both sides of the equation by 3 to solve for x.
x = -6/3
x = -2
So, the solution to the equation log4 (x-2) - log4 (x+1) = 1 is x = -2.