Solve each equation. log_1/3 (x+6) = -2

Verified step by step guidance

1. Recall the definition of a logarithm: if \(\log_{a}(b) = c\), then it is equivalent to the exponential form \(a^{c} = b\).
2. Rewrite the given equation \(\log_{\frac{1}{3}}(x+6) = -2\) in exponential form using the base \(\frac{1}{3}\) and the exponent \(-2\): \(\left(\frac{1}{3}\right)^{-2} = x + 6\).
3. Simplify the expression \(\left(\frac{1}{3}\right)^{-2}\) by applying the negative exponent rule: \(\left(\frac{1}{3}\right)^{-2} = 3^{2}\).
4. Calculate \$3^{2}\( to get the value on the right side of the equation, then set it equal to \)x + 6$.
5. Solve for \(x\) by subtracting 6 from both sides: \(x = 3^{2} - 6\).

Key Concepts

Logarithmic Functions and Their Properties

A logarithmic function is the inverse of an exponential function. The expression log_b(a) = c means that b raised to the power c equals a. Understanding this relationship allows you to rewrite logarithmic equations in exponential form to solve for the variable.

Change of Base and Negative Logarithms

Logarithms with bases between 0 and 1, such as 1/3, produce negative values for inputs greater than 1. Recognizing how the base affects the sign and behavior of the logarithm is essential when solving equations involving fractional bases.

Solving Logarithmic Equations by Exponentiation

To solve equations like log_b(x + 6) = -2, rewrite the equation in exponential form: x + 6 = b^{-2}. Then calculate the power and isolate x. This method transforms the logarithmic equation into a simpler algebraic equation.