Final answer:
To solve the equation 2log_5(x-2) + log_5 5 = 3, we rewrite it using logarithm properties and simplify to find x = 7, after ruling out an invalid solution.
Step-by-step explanation:
To solve the logarithmic equation 2log5(x-2) + log55 = 3, we can utilize the properties of logarithms to simplify the equation. Using the fact 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 2log5(x-2) as log5(x-2)2. Additionally, since log55 equals 1 (any logarithm of a number to its base equals 1), we can simplify the equation to log5(x-2)2 = 2.
Now, we can raise 5 to both sides of the equation to remove the logarithm and obtain (x-2)2 = 52. Simplifying further gives us (x-2)2 = 25. Taking the square root of both sides yields two possible solutions: x-2 = 5 or x-2 = -5. Solving for x, we get x = 7 or x = -3. However, since the domain of a logarithm function does not include negative numbers, x = -3 is not a valid solution. Therefore, the solution to the equation is x = 7.