Solve each equation. Give solutions in exact form. See Examples 5–9. log₅ [(3x + 5)(x + 1)] = 1

Verified step by step guidance

1. Recall the definition of logarithm: if \(\log_b A = C\), then \(A = b^C\). Here, the base is 5, so rewrite the equation \(\log_5 \left[(3x + 5)(x + 1)\right] = 1\) as \((3x + 5)(x + 1) = 5^1\).
2. Simplify the right side: \$5^1 = 5\(, so the equation becomes \)(3x + 5)(x + 1) = 5$.
3. Expand the left side using the distributive property: multiply \$3x\( by \)x\( and \)1\(, then multiply \)5\( by \)x\( and \)1\(, giving \)3x^2 + 3x + 5x + 5 = 5$.
4. Combine like terms on the left side: \$3x^2 + 8x + 5 = 5$.
5. Subtract 5 from both sides to set the equation to zero: \$3x^2 + 8x + 5 - 5 = 0\(, which simplifies to \)3x^2 + 8x = 0\(. Then solve this quadratic equation for \)x$.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule log_b(MN) = log_b(M) + log_b(N), is essential for simplifying and solving logarithmic equations. These properties allow you to combine or break down logarithmic expressions to isolate the variable.

Definition of Logarithm

The definition of a logarithm states that if log_b(A) = C, then b^C = A. This concept is crucial for converting logarithmic equations into exponential form, which often makes it easier to solve for the variable.

Solving Quadratic Equations

After rewriting the logarithmic equation in exponential form, you may obtain a quadratic equation. Knowing how to solve quadratic equations using factoring, completing the square, or the quadratic formula is necessary to find the exact solutions.