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

Verified step by step guidance

1. Recall the definition of logarithm: if \(\log_{a}(b) = c\), then it means \(a^{c} = b\). Here, the base is 5 and the logarithm equals 1, so rewrite the equation \(\log_{5} \left[(3x + 5)(x + 1)\right] = 1\) as an exponential equation: \$5^{1} = (3x + 5)(x + 1)$.
2. Simplify the right-hand side by expanding the product: multiply \((3x + 5)\) by \((x + 1)\) using the distributive property (FOIL method): \((3x)(x) + (3x)(1) + 5(x) + 5(1)\).
3. Write the expanded expression as a quadratic equation: \$3x^{2} + 3x + 5x + 5 = 5\(, then combine like terms to get \)3x^{2} + 8x + 5 = 5$.
4. Subtract 5 from both sides to set the quadratic equation equal to zero: \$3x^{2} + 8x + 5 - 5 = 0\(, which simplifies to \)3x^{2} + 8x = 0$.
5. Solve the quadratic equation \$3x^{2} + 8x = 0\( by factoring out the common factor \)x\(: \)x(3x + 8) = 0\(. Then set each factor equal to zero and solve for \)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. This allows the expression log_5[(3x + 5)(x + 1)] to be expanded or manipulated to simplify solving the equation.

Definition of Logarithms and Exponentials

The definition log_b(A) = C means b^C = A. Applying this definition helps convert the logarithmic equation into an exponential form, making it easier to solve for x by removing the logarithm.

Solving Quadratic Equations

After rewriting the equation in polynomial form, solving the resulting quadratic equation is necessary. Techniques include factoring, completing the square, or using the quadratic formula to find exact solutions for x.