Question

Evaluate the expression without using a calculator.\(\ln e^{-4}\)

Short answer

The evaluated expression \(\ln e^{-4}\) is -4.

Step-by-step solution

Identify the given expression

The given expression to evaluate is \(\ln e^{-4}\).

Apply the law of logarithms

\(\ln e^{a} = a\). In the given expression, \[ a = -4 \]. Thus, applying the law of logarithms we get the solution without needing a calculator.

Key concepts

Laws of Logarithms

Understanding the laws of logarithms is crucial for simplifying and solving logarithmic expressions. These laws are based on the fundamental properties of logarithms, which allow us to manipulate the expressions to make calculations easier.

One important law is the power rule, which states that the logarithm of a power is equal to the exponent times the logarithm of the base, mathematically represented as \(\log_b{a^n} = n\cdot\log_b{a}\). This is particularly useful when dealing with natural logarithms, where the base is the mathematical constant \(e\). For instance, when we have an expression like \(\ln e^{-4}\), we can apply this law to directly extract the exponent, resulting in the simple evaluation of \(-4\), without the need for a calculator.

Other laws include the product rule, stating that the logarithm of a product equals the sum of the logarithms of the factors, and the quotient rule, which tells us that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. By mastering these rules, complex logarithmic expressions become far more manageable.

Natural Logarithm Properties

The natural logarithm, denoted as \(\ln\), is a logarithm that has the number \(e\), approximately equal to 2.718, as its base. Its unique properties make it a fundamental tool in various branches of mathematics and science.

One key property of the natural logarithm is its relationship to the number \(e\), expressed through the identity \(\ln e = 1\). This stems from the definition that \(e\) is the unique number for which the area under the curve \(y = 1/x\) from 1 to \(e\) is equal to 1.

Another significant property is \(\ln e^x = x\), which we used in the given exercise. This property simplifies the process of evaluating expressions involving \(e\) raised to any power, which is quite useful in both calculus and algebra. In addition, the natural logarithm function is the inverse operation of the exponential function with base \(e\), meaning that \(\ln(e^x) = x\) and \(e^{\ln x} = x\) for all positive values of \(x\). This inverse relationship is essential for solving equations involving exponential and logarithmic terms.

Exponential Functions

Exponential functions, which can be written in the form \(f(x) = e^x\), where \(e\) is Euler's number, are fundamental in mathematics due to their unique characteristics and applications.

One of the defining qualities of an exponential function is its rate of growth. Unlike linear functions that grow at a constant rate, the rate of growth of an exponential function is proportional to its current value, which leads to rapidly increasing quantities. This behavior is best illustrated in phenomena such as compounding interest, population growth, and certain natural processes.

Another key aspect is the derivative and integral of an exponential function, which are remarkably simple relative to other functions. The derivative of \(e^x\) with respect to \(x\) is itself, \(\frac{d}{dx}e^x = e^x\), while the integral of \(e^x\) over \(x\) is also \(e^x\) (plus the constant of integration). This self-replicating property is part of what defines \(e\) and makes it a natural choice for the base of natural logarithms.

Exponential functions with base \(e\) are also essential in describing decay processes and growth scenarios, such as in physics and biology. Their ubiquitous applications and ties to logarithms highlight the intertwined nature of these mathematical concepts.