Solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0

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Recall the logarithm property that allows combining sums and differences of logarithms: \(\ln a + \ln b = \ln(ab)\) and \(k \ln a = \ln(a^k)\). Use these to combine the terms on the left side of the equation.

Apply the properties to rewrite the equation \(\ln(2x+1) + \ln(x-3) - 2 \ln x = 0\) as a single logarithm: \(\ln\left((2x+1)(x-3)\right) - \ln(x^2) = 0\).

Use the logarithm subtraction property \(\ln A - \ln B = \ln\left(\frac{A}{B}\right)\) to combine the expression into one logarithm: \(\ln\left(\frac{(2x+1)(x-3)}{x^2}\right) = 0\).

Since \(\ln y = 0\) implies \(y = 1\), set the argument of the logarithm equal to 1: \(\frac{(2x+1)(x-3)}{x^2} = 1\).

Multiply both sides by \(x^2\) to clear the denominator and then expand and simplify the resulting quadratic equation. Solve for \(x\), remembering to check that your solutions satisfy the original domain restrictions for the logarithms.

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Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. For example, ln(a) + ln(b) = ln(ab) and k ln(a) = ln(a^k). These allow combining or simplifying logarithmic expressions to solve equations effectively.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments. When solving equations involving ln(x), ensure that expressions inside the logarithms are greater than zero to find valid solutions and avoid extraneous roots.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, then exponentiating both sides to eliminate the logarithm. This transforms the equation into a polynomial or rational form that can be solved using algebraic methods.