Final answer:
To solve for x in log₁ᵡ(3x²-10x+12)=2, we rewrite the equation as x² = 3x²-10x+12, and rearrange it into a quadratic equation 2x² - 10x + 12 = 0, which can be solved using the quadratic formula. Solutions that result in a logarithm base of zero, one, or a negative number must be disregarded.
Step-by-step explanation:
The question asks us to solve for all values of x in the equation log₁ᵡ(3x²-10x+12)=2. We'll use the property of logarithms that the log of a number y to the base x equals n can be written as xn = y. This gives us x2 = 3x²-10x+12. Now we bring all terms to one side of the equation to form a quadratic equation: 3x² - 10x + 12 - x² = 0, which simplifies to 2x² - 10x + 12 = 0. This quadratic equation can be solved using the quadratic formula, x = √[b² - 4ac] / 2a.
However, since we cannot have a logarithm base of zero or one, and the base cannot be negative, solutions violating these restrictions must be discarded.