In Exercises 74–79, solve each logarithmic equation. log₄ (2x+1) = log₄ (x-3) + log₄ (x+5)

Verified step by step guidance

1. Recall the logarithmic property that allows you to combine the sum of logarithms with the same base: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to the right side of the equation to combine the two logarithms into one.
2. Rewrite the equation using the property: \(\log_4 (2x + 1) = \log_4 ((x - 3)(x + 5))\).
3. Since the logarithms on both sides have the same base and are equal, set their arguments equal to each other: \$2x + 1 = (x - 3)(x + 5)$.
4. Expand the right side by multiplying the binomials: \((x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15\).
5. Set up the equation \$2x + 1 = x^2 + 2x - 15\( and then rearrange all terms to one side to form a quadratic equation: \)0 = x^2 + 2x - 15 - 2x - 1\(, which simplifies to \)0 = x^2 - 16\(. From here, solve for \)x$.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. In this problem, the product rule allows combining the sum of logarithms on one side into a single logarithm, simplifying the equation for easier solving.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties and then converting it to an exponential form or equating the arguments of the logarithms when the bases are the same. This approach helps isolate the variable and find its value.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations, it is crucial to check that the solutions satisfy the domain restrictions (arguments inside the logs must be greater than zero) to ensure valid answers.