In Exercises 93–102, solve each equation. ln(2x+1)+ln(x−3)−2 ln x=0

Verified step by step guidance

1. Recognize that the equation involves natural logarithms and use the properties of logarithms to combine them. Recall that $\ln (a)+\ln (b)=\ln (ab)$ and $\ln (a)-\ln (b)=\ln \left(\frac{a}{b}\right)$.
2. Combine the logarithms on the left side: $\ln ((2x+1)(x-3))-\ln (x^2)=0$.
3. Use the property $\ln (a)-\ln (b)=\ln \left(\frac{a}{b}\right)$ to rewrite the equation as $\ln \left(\frac{(2x+1)(x-3)}{x^2}\right)=0$.
4. Since $\ln (y)=0$ implies $y=1$, set the argument of the logarithm equal to 1: $\frac{(2x+1)(x-3)}{x^2}=1$.
5. Solve the resulting equation $(2x+1)(x-3)=x^2$ by expanding and simplifying to find the values of $x$.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying logarithmic expressions. Key properties include the product rule (ln(a) + ln(b) = ln(ab)), the quotient rule (ln(a) - ln(b) = ln(a/b)), and the power rule (k * ln(a) = ln(a^k)). These properties allow us to combine and manipulate logarithmic terms effectively in equations.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and algebra due to its unique properties, such as the fact that the derivative of ln(x) is 1/x. Understanding how to work with ln is crucial for solving equations involving exponential growth or decay.

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved by rewriting them in logarithmic form. For instance, if we have an equation like e^y = x, we can take the natural logarithm of both sides to solve for y: y = ln(x). Recognizing how to transition between exponential and logarithmic forms is vital for solving equations that include ln.