Solve each equation. x = log₃ 1/81

Verified step by step guidance

1. Recognize that the equation is given in logarithmic form: \(x = \log_{3} \frac{1}{81}\). This means \(x\) is the exponent to which the base 3 must be raised to get \(\frac{1}{81}\).
2. Rewrite the equation in its equivalent exponential form: \(3^{x} = \frac{1}{81}\).
3. Express \(\frac{1}{81}\) as a power of 3. Since \$81 = 3^{4}$, then \(\frac{1}{81} = 3^{-4}\).
4. Set the exponents equal to each other because the bases are the same: \$3^{x} = 3^{-4}\( implies \)x = -4$.
5. Conclude that the solution to the equation is \(x = -4\).

Key Concepts

Logarithms and Their Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₃(1/81) asks for the power to which 3 must be raised to get 1/81. Understanding this definition is essential to solving logarithmic equations.

Properties of Exponents

Since logarithms are inverses of exponents, knowing how to manipulate exponents is crucial. Recognizing that 81 is 3 raised to the 4th power (3⁴ = 81) helps rewrite 1/81 as 3⁻⁴, simplifying the logarithmic expression.

Solving Logarithmic Equations

To solve equations like x = log₃(1/81), rewrite the logarithmic form into exponential form: 3ˣ = 1/81. Then, express 1/81 as a power of 3 and solve for x by equating exponents, which leads to the solution.