Step 1: Simplify the given equation. The equation is:
log3(x − 2) = 2log25(2x − 4)
Using the property a logb(c) = logb(ca), rewrite the equation:
log3(x − 2) = log25(2x − 4)2
Step 2: Convert the logarithmic bases to the same base. Using logb(a) = $\frac{\log(a)}{\log(b)}$, the equation becomes:
$\frac{\log(x - 2)}{\log(3)} = \frac{\log((2x - 4)^2)}{\log(25)}$
Simplify further:
log(x − 2) ⋅ log(25) = log(3) ⋅ log((2x − 4)2)
Step 3: Solve for x. Expand log((2x − 4)2) using log(ab) = b log(a):
log(x − 2) ⋅ log(25) = 2log(3) ⋅ log(2x − 4)
Let u = log(x − 2) and v = log(2x − 4). Substitute:
u ⋅ log(25) = 2v ⋅ log(3)
After solving, x = 3 and x = 5. Therefore, the equation has 2 solutions.
Answer: 2