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. log₄(3x+2)=3

Verified step by step guidance

1. Recall the definition of logarithm: if \(\log_{a}(b) = c\), then it is equivalent to the exponential form \(a^{c} = b\).
2. Rewrite the given equation \(\log_{4}(3x + 2) = 3\) in exponential form: \$4^{3} = 3x + 2$.
3. Calculate \$4^{3}\( (but do not finalize the numeric value) and set up the equation: \)64 = 3x + 2$.
4. Solve for \(x\) by isolating it: subtract 2 from both sides to get \$64 - 2 = 3x\(, then divide both sides by 3 to find \)x = \frac{62}{3}$.
5. Check the domain restriction: the argument of the logarithm, \$3x + 2\(, must be greater than 0. Substitute \)x = \frac{62}{3}\( to verify \)3\left(\frac{62}{3}\right) + 2 > 0$ to ensure 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 you to rewrite logarithmic equations in exponential form to solve for the variable. Recognizing how to manipulate logs helps simplify and solve the given equation.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all values for which the argument (inside the log) is positive. For log_4(3x+2), the expression 3x+2 must be greater than zero. Identifying and applying domain restrictions ensures that solutions are valid and prevents extraneous answers.

Exact and Approximate Solutions

After solving the equation exactly, it is often necessary to provide a decimal approximation. Using a calculator to find decimal values correct to two decimal places helps interpret the solution practically. Distinguishing between exact and approximate answers is important in many algebra problems.