Final answer:
The solution to the logarithmic equation log₃(x-2)²=4 is x=11, which is found by using properties of logarithms and converting to exponential form.
Step-by-step explanation:
The solution to the logarithmic equation log₃(x-2)²=4 can be found using logarithmic and exponential operations. By applying the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we can rewrite the equation as 2·log₃(x-2)=4. Then, we simplify this to log₃(x-2)=2. By converting from logarithmic form to exponential form, we get (x-2) = 3², which simplifies to x-2 = 9. Finally, adding 2 to both sides, we find that x = 11.