Final answer:
To solve the equation log(4)(2x+1) = log(4)(x+2) - log(4)3, we apply logarithmic properties to simplify and combine terms, which then allows us to solve for x algebraically, finding that x equals -1/5.
Step-by-step explanation:
The question is asking to solve the logarithmic equation log(4)(2x+1) = log(4)(x+2) - log(4)3 using properties of logarithms. To solve this equation, we can apply the property that states the logarithm of the quotient of two numbers is the difference of their logarithms. Applying this property, we can combine the right side into a single logarithm. So, log(4)(x+2) - log(4)3 becomes log(4)((x+2)/3).
Next, we equate the arguments of the logarithms because if two logs of the same base are equal, their arguments must also be equal. This gives us the equation 2x+1 = (x+2)/3. Multiplying both sides by 3 to clear the fraction yields 6x + 3 = x + 2. Solving for x, we subtract x from both sides and get 5x + 3 = 2. Finally, subtracting 3 from both sides and dividing by 5, we find that x = -1/5.