In Exercises 93–102, solve each equation. ln 3−ln(x+5)−ln x=0

Verified step by step guidance

1. Recognize that the equation involves natural logarithms and can be simplified using logarithmic properties.
2. Apply the property of logarithms: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). This allows you to combine the logarithms: \( \ln 3 - \ln(x+5) - \ln x = \ln \left( \frac{3}{(x+5)x} \right) \).
3. Set the simplified logarithmic expression equal to zero: \( \ln \left( \frac{3}{(x+5)x} \right) = 0 \).
4. Recall that if \( \ln y = 0 \), then \( y = 1 \). Therefore, set the argument of the logarithm equal to 1: \( \frac{3}{(x+5)x} = 1 \).
5. Solve the resulting equation \( \frac{3}{(x+5)x} = 1 \) for \( x \) by cross-multiplying and simplifying the expression.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify 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)). Understanding these properties is essential for manipulating logarithmic equations effectively.

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. Recognizing how to work with ln is crucial for solving equations involving logarithms.

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved by rewriting them in logarithmic form. For instance, if a^x = b, then x = log_a(b). Understanding how to convert between exponential and logarithmic forms is vital for solving equations that include logarithmic expressions.