Final answer:
To solve log³(x+2) + log³(6) = 3, use the property that the logarithm of a product is the sum of the logarithms of the factors to find x as 2.5.
Step-by-step explanation:
The question involves solving the equation log³(x+2) + log³(6) = 3 for x, where log³ refers to the logarithm with base 3. To solve this, we can use the properties of logarithms, particularly the property that the logarithm of a product of two numbers is the sum of the logarithms of the two numbers (log xy = log x + log y).
Applying this property to combine the two logarithms, we get log³((x+2)*6) = 3.
Next, to find the value inside the log function, we raise 3 to the power of 3, which gives us 27.
Equating this to our combined expression (x+2)*6, we can solve for x: (x+2)*6 = 27.
Dividing both sides of the equation by 6 gives x+2 = 27 / 6, which simplifies to x+2 = 4.5.
Finally, subtracting 2 from both sides of the equation yields x = 2.5.