Solve each equation. log￬1/2 (x+3) = -4

Verified step by step guidance

1. Understand that the equation \( \log_{1/2}(x+3) = -4 \) is in logarithmic form, where the base is \( \frac{1}{2} \), the argument is \( x+3 \), and the result is \(-4\).
2. Convert the logarithmic equation to its equivalent exponential form: \( x+3 = (\frac{1}{2})^{-4} \).
3. Recognize that raising a fraction to a negative exponent involves taking the reciprocal and then raising it to the positive exponent: \( (\frac{1}{2})^{-4} = 2^4 \).
4. Calculate \( 2^4 \) to simplify the equation: \( x+3 = 16 \).
5. Solve for \( x \) by subtracting 3 from both sides of the equation: \( x = 16 - 3 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They express the power to which a base must be raised to obtain a certain value. In the equation log_b(a) = c, b is the base, a is the argument, and c is the exponent. Understanding how to manipulate and solve logarithmic equations is essential for solving problems involving logarithms.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, which can simplify calculations. It states that log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms that are not easily computable in their original base, enabling the use of common or natural logarithms instead.

Exponential Equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. To solve logarithmic equations, one often converts them into exponential form. For example, if log_b(a) = c, it can be rewritten as b^c = a. This transformation is crucial for isolating the variable and finding its value in logarithmic problems.