91Ó°ÊÓ

When solving equations involving absolute value, the first step is to isolate the absolute value itself. Absolute value represents the distance a number is from zero on the number line, regardless of direction. In this context, \( |\ln x| \) is the absolute value of the logarithm of \( x \). Since the absolute value can represent both a positive and a negative scenario, you'll have two possible equations to solve. These are derived from setting \( \ln x \) equal to both the positive and negative values of the isolated number. In our equation, \( 2|\ln x|-6=0 \), isolating leads to \( |\ln x|=3 \). Thus, the two scenarios are \( \ln x = 3 \) and \( \ln x = -3 \). This provides a broader set of potential solutions.

Logarithmic functions are the inverse of exponential functions. The equation \( \ln x \) signifies the natural logarithm of \( x \), which is the power to which \( e \) (approximately 2.718) must be raised to produce \( x \). In simpler terms, if \( \ln x = 3 \), it means \( e^3 = x \). Logarithmic equations are crucial for solving for \( x \) in situations where \( x \) is an exponent itself. They transform multiplicative relationships into additive ones, simplifying the process of finding unknowns in equations. By addressing \( \ln x = 3 \) and \( \ln x = -3 \), we efficiently find two solutions using exponential equivalences: \( x = e^3 \) and \( x = e^{-3} \). This illustrates the transformative power of logarithms in mathematics.

The properties of logarithms make solving equations with logarithmic expressions manageable and systematic. Key properties to remember include:

• The logarithm of a power: \( \log_b(a^n) = n \log_b(a) \)
• The exponential log equivalence: \( b^{\log_b(y)} = y \)

These properties allow you to switch between logarithmic and exponential forms seamlessly. In our exercise, once \( |\ln x| = 3 \) is set, translating \( \ln x = 3 \) into \( x = e^3 \) utilizes the equivalence between logarithmic and exponential operations. Recognizing these properties enables a student to break down complex expressions and equations, paving a straightforward path to finding solutions. By using these properties carefully, equations that initially seem daunting break down into smaller, more manageable parts.