Question

Order the steps to solve the equation log₃(x+2)=log_(3)(2x²-1) from 1 to 6 0=(2x-3)(x+1) 0=2x²-x-3 Potential solutions are -1 and (3)/(2). 2x-3=0 or x+1=0 x+2=2x²-1 ³(log₃(x+2))=³(log₃(2x²-1))

Answer 1

To solve the logarithmic equation, set the arguments of the logarithms with the same base equal, rearrange to form a quadratic equation, factor, solve for x, and check the solutions.

Step-by-step explanation:

To solve the equation log₃(x+2)=log₃(2x²-1), we need to follow a series of steps, and utilizing the properties of logarithms is key here. Since the bases of the logarithms are the same, we can set the arguments equal to each other and solve the resulting equation. Here are the ordered steps:

1. Given that the bases are the same, we can equate the arguments: x+2 = 2x² - 1.
2. Rearrange the equation to set it to zero: 0 = 2x² - x - 3.
3. Factor the quadratic equation: 0 = (2x - 3)(x + 1).
4. Set each factor equal to zero: 2x - 3 = 0 or x + 1 = 0.
5. Solve each equation for x: potential solutions are x = ⅓ and x = -1.
6. Verify the solutions in the original equation.

The third step, invoking the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, is not directly used in this problem. However, this knowledge is essential in understanding logarithms and could be applied in different scenarios involving the manipulation of exponential functions.