Answer: To solve the equation log3x - log3(x - 8) = 2, we can use the properties of logarithms to simplify and solve for x.
First, let's apply the quotient rule of logarithms. The quotient rule states that log(base a)(b) - log(base a)(c) = log(base a)(b/c).
Using this rule, we can rewrite the equation as log3(x / (x - 8)) = 2.
Next, let's rewrite 2 as a logarithm. The logarithmic form of 2 is log(base a)(b) = c, where a^c = b. In this case, a^c = 3^2 = 9. Therefore, we can rewrite the equation as log3(x / (x - 8)) = log3(9).
Now that the bases are the same, we can set the arguments of the logarithms equal to each other. Therefore, x / (x - 8) = 9.
To solve for x, we can multiply both sides of the equation by (x - 8) to eliminate the fraction. This gives us x = 9(x - 8).
Expanding the right side of the equation, we get x = 9x - 72.
Next, we can subtract 9x from both sides of the equation to isolate x. This gives us -8x = -72.
Dividing both sides of the equation by -8, we find that x = 9.
Therefore, the solution to the equation log3x - log3(x - 8) = 2 is x = 9.
Explanation: