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. 2 log3(x+4)=log3 9+2

Verified step by step guidance

1. Start by using the properties of logarithms to simplify the equation: $2{\log }_3(x+4)={\log }_{3}9+2$.
2. Recognize that $2$ on the right side can be rewritten as ${\log }_3(3^2)$, so the equation becomes $2{\log }_3(x+4)={\log }_{3}9+{\log }_{3}9$.
3. Apply the property $a{\log }_{b}c={\log }_b(c^a)$ to the left side: ${\log }_3((x+4{)}^2)={\log }_3(9\times 9)$.
4. Since the bases of the logarithms are the same, set the arguments equal to each other: $(x+4{)}^2=81$.
5. Solve the equation $(x+4{)}^2=81$ by taking the square root of both sides, remembering to consider both the positive and negative roots.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The logarithm log_b(a) answers the question: 'To what power must the base b be raised to obtain a?' Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating and solving logarithmic equations.

Domain of Logarithmic Expressions

The domain of a logarithmic expression is the set of all input values (x) for which the logarithm is defined. For log_b(x), x must be greater than zero (x > 0). When solving logarithmic equations, it is crucial to check the solutions against the original expressions to ensure they fall within the valid domain, as extraneous solutions may arise during the solving process.

Solving Logarithmic Equations

To solve logarithmic equations, one typically uses properties of logarithms to combine or simplify the expressions. This may involve rewriting the equation in exponential form or applying logarithmic identities. After finding potential solutions, it is important to verify them by substituting back into the original equation to ensure they are valid within the defined domain.