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. Identify the given logarithmic equation: $\log 3_{}(x+4)=-3$.
2. Recall the definition of logarithm: $\log {\mathrm{a}}_{}\left(b\right)=c$ means $a^c=b$. Apply this to rewrite the equation in exponential form: $3^{-}=x+4$.
3. Simplify the exponential expression: $3^{-}=\frac{1}{3^3}=\frac{1}{27}$. So, the equation becomes $\frac{1}{27}=x+4$.
4. Solve for $x$ by isolating it: $x=\frac{1}{27}-4$.
5. Check the domain of the original logarithmic expression: since $log\_3(x+4)$ is defined only if $x\; +\; 4\; >\; 0$, verify that the solution satisfies $x\; >\; -4$. If it does, the solution is valid.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows converting the logarithmic equation into an exponential form to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires the argument x to be positive. When solving equations, any solution that makes the argument non-positive must be rejected to ensure the solution is valid.

Exact and Approximate Solutions

After finding the exact solution, it is often necessary to provide a decimal approximation using a calculator. This helps interpret the solution practically, especially when the exact form is a fraction or irrational number.