Final answer:
To solve the logarithmic equation log_{1/2}(x+6) = -3, we convert it to exponential form to find (1/2)^{-3} = x + 6. Then we calculate that 2^3 equals 8, and by subtracting 6 from both sides, we discover that x equals 2.
Step-by-step explanation:
To solve the given logarithmic equation log_{1/2}(x+6) = -3, we need to understand the definition of a logarithm. A logarithm log_b(a) is the power to which we have to raise the base 'b' in order to get 'a'. In this case, our base is '1/2' and our power is '-3'. From the logarithmic form, we can convert to exponential form to find the value of 'x'.
Converting the equation to exponential form gives us: (1/2)^{-3} = x + 6. Calculating (1/2)^{-3}, we get 8 because negative exponents mean that the base is taken as the reciprocal, and then raised to the absolute value of the exponent: (2/1)^3 which is 2^3 = 8. Now we can solve for 'x' by subtracting 6 from both sides.
x + 6 = 8
x = 8 - 6
x = 2
The solution to the logarithmic equation is x = 2.