Question

Log₃(x²+3x+3)=log₃(x+2)³ +3x+3

Answer 1

To solve the equation log₃(x²+3x+3)=log₃(x+2)³ +3x+3, you can use logarithmic properties. First, apply the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. This allows you to simplify the equation and then solve for x.

Step-by-step explanation:

To solve the equation log₃(x²+3x+3)=log₃(x+2)³ +3x+3, we can use logarithmic properties. First, apply the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. This allows us to simplify the equation to x²+3x+3 = (x+2)³(x+2)³ +3x+3. Then, expand and simplify the equation to solve for x.

Here's the step-by-step process to solve the equation:

1. Apply the logarithmic property: x²+3x+3 = (x+2)³(x+2)³ +3x+3
2. Expand the equation: x²+3x+3 = (x+2)(x+2)(x+2)(x+2) +3x+3
3. Simplify the equation: x²+3x+3 = (x+2)⁴ +3x+3
4. Expand the equation further and combine like terms: x²+3x+3 = x⁴ + 4x³ + 4x² + 8x + 8 + 3x + 3
5. Rearrange the equation to bring all terms to one side: x⁴ + 4x³ - x² - 2x - 5 = 0
6. Factor the equation if possible or use numerical methods to find the solutions for x.