Final answer:
The given logarithmic equation is transformed into a quadratic equation, which after further investigation with the discriminant, has no real solution, making the correct answer (e) No solution.
Step-by-step explanation:
The logarithmic equation log₅(x²−6x+8) equals log₅(−8x) implies that the expressions inside the logarithms must be equal since the bases are the same and the logarithmic function is one-to-one. This means that x² - 6x + 8 must be equal to −8x. Therefore, we can set up the equation x² - 6x + 8 = −8x. Combining like terms, we get x² + 2x + 8 = 0.
To find the solutions to this quadratic equation, we can use the quadratic formula. However, upon trying to factor, we find that the quadratic does not factor nicely, so we are suspecting that there may be non-integer roots or no real roots at all. Calculating the discriminant (Δ = b² - 4ac), we have (2)² - 4(1)(8) = 4 - 32 = -28, which is negative. Since the discriminant is negative, this equation has no real solution.
Therefore, the correct answer is (e) No solution.