91Ó°ÊÓ

Exponential functions form a fundamental part of mathematics, especially important for describing growth and decay. An exponential function is typically expressed in the form \(a^x\), where \(a\) is a constant and \(x\) is the variable. A special case of exponential functions is \(e^x\), where \(e\) is the base of the natural logarithm, approximately valued at 2.718.

• Exponential growth occurs when the base \(a > 1\), causing the function to rapidly increase as \(x\) gets larger.
• Exponential decay takes place when \(0 < a < 1\), which causes the function to decrease as \(x\) grows.

In the context of our exercise, we deal with \(e^{-2x}\). Since the natural logarithm and exponentials are inverses, \(\ln(e^{-2x})\) simplifies directly to \(-2x\). This inverse relationship is crucial for various applications in both mathematics and real-world problems, such as calculus and natural phenomena modeling.

Understanding logarithmic identities is essential for simplifying expressions, like the one in our exercise. The natural logarithm \(\ln\) is the logarithm with base \(e\). Some important identities are often used to make calculations more manageable:

• Identity for powers: \(\ln(a^b) = b\ln(a)\).
• Inverse identification: \(\ln(e^a) = a\).
• Logarithm of one: \(\ln(1) = 0\)

In the given exercise, we actually used two key identities. First, the inverse property helped us simplify \(\ln(e^{-2x})\) to \(-2x\). Next, the logarithm of one property allowed us to recognize \(\ln 1\) as zero, which further streamlined the expression. Understanding these identities enables one to manipulate logarithmic expressions fluently, making solving logarithmic equations much simpler.

Function simplification is about reducing a mathematical expression to its simplest and most concise form. It helps to understand the behavior of the function quickly and makes further calculations more efficient. The original function given was:\[f(x) = \ln(e^{-2x}) + 3x + \ln 1\]By applying known properties step-by-step, we simplified it as follows:

• First, using the inverse relationship \(\ln(e^{\text{something}}) = \text{something}\), simplified to \(-2x\).
• Identify that \(\ln 1\) equals 0, removing it from the expression completely.
• Combine like terms \(-2x + 3x\) to produce a final result of \(x\).

The simplified function \(f(x) = x\) both represents the original more directly and is easier to work with for further analysis, calculations, or graphing. Simplifying functions is an essential skill in mathematics, enabling more straightforward insights and processing.