Solve each equation. ln 3−ln(x+5)−ln x=0

Verified step by step guidance

1. Recall the logarithmic property that allows you to combine the difference of logarithms: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\). Use this to combine the terms on the left side of the equation.
2. Apply the property to rewrite \(\ln 3 - \ln (x+5) - \ln x\) as \(\ln \left( \frac{3}{(x+5) \cdot x} \right)\).
3. Set the equation \(\ln \left( \frac{3}{x(x+5)} \right) = 0\) and recall that \(\ln A = 0\) implies \(A = 1\).
4. From the previous step, write the equation \(\frac{3}{x(x+5)} = 1\) and multiply both sides by \(x(x+5)\) to clear the denominator.
5. Simplify the resulting equation to a quadratic form: \$3 = x(x+5)\(, which expands to \)3 = x^2 + 5x\(. Rearrange to standard quadratic form \)x^2 + 5x - 3 = 0\( and prepare to solve for \)x$ using the quadratic formula.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. For example, the difference of logarithms, ln a - ln b, can be rewritten as ln(a/b). This allows simplification of expressions involving multiple logarithms into a single logarithm.

Solving Logarithmic Equations

Solving logarithmic equations involves isolating the logarithmic expression and then rewriting the equation in exponential form. This step helps to eliminate the logarithm and solve for the variable. Checking for extraneous solutions is important since the domain of logarithms is restricted to positive arguments.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations like ln(x+5) or ln x, the expressions inside the logarithms must be greater than zero. Identifying and applying these domain restrictions ensures that solutions are valid and meaningful.