Solve the following equation:log2(6𝑥)−log2(𝑥−8)=4

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

The equation is: log2(6x) - log2(x-8) = 4

Step 1: Use the properties of logarithms to combine the two logarithms on the left side of the equation. The difference of two logarithms is the logarithm of the quotient, so we can write:

log2[(6x)/(x-8)] = 4

Step 2: To get rid of the logarithm, we can use the property of logarithms that says that if logb(a) = c, then b^c = a. In this case, b is 2, a is (6x)/(x-8), and c is 4. So we can write:

(6x)/(x-8) = 2^4

Step 3: Simplify the right side of the equation:

(6x)/(x-8) = 16

Step 4: Multiply both sides of the equation by (x-8) to get rid of the denominator on the left side:

6x = 16(x-8)

Step 5: Distribute the 16 on the right side:

6x = 16x - 128

Step 6: Subtract 16x from both sides to get the x terms on one side:

-10x = -128

Step 7: Divide both sides by -10 to solve for x:

x = 128/10 = 12.8

So the solution to the equation is x = 12.8.