Final answer:
To find the largest value of x that satisfies the equation log₅(x²) − log₅(x+2) = 5, we use logarithmic properties to simplify and convert it into a quadratic equation. After applying the quadratic formula, we carefully consider the domain restrictions to determine the largest valid solution for x.
Step-by-step explanation:
Solving the Logarithmic Equation
The student has asked us to find the largest value of x that satisfies the equation log₅(x²) − log₅(x+2) = 5. The properties of logarithms can be applied to simplify and solve this equation. We use the property that log(a) - log(b) = log(a/b), which is valid for any base of the logarithm, including base 5, as in this problem.
First, we combine the logarithmic expressions on the left side using the aforementioned property: log₅(x² / (x+2)) = 5
Then, we can rewrite the equation in exponential form since if log₅(A) = B, then 5^B = A: 5^5 = x² / (x+2)
Multiply both sides by (x+2) to isolate the x² term: 5^5 * (x+2) = x²
Now, solve the resulting quadratic equation for x. We can either factor the equation or use the quadratic formula. Factoring might require testing possible factors, and if the equation does not factor easily, recourse to the quadratic formula is the best option. Remember that the quadratic formula is x = (-b ± √(b²-4ac)) / (2a), where a, b, and c are the coefficients in the quadratic equation ax² + bx + c = 0. Finally, we check the possible solutions because only valid solutions are those that make the argument of both logs positive, so x must be greater than zero and not equal to -2. To get the exact largest value of x, we’d continue by solving the quadratic, ensuring we only consider the valid solutions.