solve the equation
log2(3x-2) - log2 (x-5)=4
3x-2/x-5 = 2^4
3x-2/x-5 = 16
multiply by x-5
x-5(3x-2/x-5) = 16(x-5)
3x -2 = 16x -80
3x -16x -2 16x-16x -80
-13x -2 = -80
-13x -2 + 2 = -80 +2
-13x = -78
x = 6
To solve the equation log2(3x-2) - log2 (x-5) = 4, we can use the properties of logarithms. The goal is to simplify the equation and isolate the variable x.
Step 1: Use the quotient rule of logarithms.
log2((3x-2)/(x-5)) = 4
Step 2: Rewrite the equation in exponential form.
2^4 = (3x-2)/(x-5)
Step 3: Simplify the left side of the equation.
16 = (3x-2)/(x-5)
Step 4: Multiply both sides of the equation by (x-5) to remove the denominator.
16(x-5) = 3x-2
Step 5: Distribute and simplify.
16x - 80 = 3x - 2
Step 6: Move all the x terms to one side and the constant terms to the other side.
16x - 3x = 2 - 80
Step 7: Simplify.
13x = -78
Step 8: Divide both sides of the equation by 13 to solve for x.
x = -78/13
Thus, the solution to the equation log2(3x-2) - log2 (x-5) = 4 is x = -6.