Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₃(x+4)=−3

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 \(\log_{3}(x+4) = -3\) in exponential form: \$3^{-3} = x + 4$.
3. Calculate \$3^{-3}\( as \)\frac{1}{3^{3}} = \frac{1}{27}\(, so the equation becomes \)\frac{1}{27} = x + 4$.
4. Solve for \(x\) by subtracting 4 from both sides: \(x = \frac{1}{27} - 4\).
5. Check the domain of the original logarithmic expression: since \(\log_{3}(x+4)\) requires \(x + 4 > 0\), verify that your solution satisfies this condition.

Key Concepts

Properties of Logarithms

Understanding the basic properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(f(x)) requires that the argument f(x) be positive. Identifying and applying this domain restriction ensures that any solution found is valid and does not produce undefined expressions.

Exact and Approximate Solutions

After solving the equation exactly, it is often necessary to find a decimal approximation using a calculator. This helps interpret the solution in a practical context, especially when the exact form is a fraction or irrational number.