Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, use the definition of common and natural logarithms to simplify. \(\ln \left(e^{-5.03}\right)\)

Step-by-Step Solution

Answer

The simplified expression is -5.03.

Step 1: Understand the problem

We are given an expression \( \ln(e^{-5.03}) \) and need to simplify it using the properties of logarithms and exponents.

Step 2: Recall the property of natural logarithms

The natural logarithm function, \( \ln(x) \), is the inverse of the exponential function with base \(e\). Thus, \( \ln(e^x) = x \) for any real number \(x\). This property will help us simplify the given expression.

Step 3: Apply the natural logarithm property

Using the property \( \ln(e^x) = x \), we can simplify \( \ln(e^{-5.03}) \) to \( -5.03 \).

Step 4: Write the simplified result

The expression \( \ln(e^{-5.03}) \) simplifies to \( -5.03 \).

Key Concepts

Natural Logarithms

Natural logarithms, denoted as \( \ln(x) \), are logarithms with base \( e \), where \( e \) is approximately equal to 2.71828. This is a special type of logarithm because the base \( e \), known as Euler's number, naturally arises in the process of continuous growth processes such as compound interest or population growth.
Natural logarithms are particularly useful in calculus and natural sciences due to their important property with the exponential function. In fact, they are the inverse of the exponential function with base \( e \). This relationship can be expressed as \( \ln(e^x) = x \) for any real number \( x \). By understanding this relationship, you can solve problems involving exponential growth or decay efficiently. For example, knowing that \( \ln(e^{-5.03}) = -5.03 \) allows us to simplify such expressions quickly by applying this property.
Remember, the natural logarithm can be used to "undo" the exponential function and vice versa. This inverse property makes it a powerful tool in simplifying equations involving exponents.

Properties of Logarithms

The properties of logarithms are essential tools for simplifying and manipulating logarithmic expressions. There are several key properties that students should familiarize themselves with. These properties make it easier to solve equations involving logs or to simplify complex expressions.

• Product Property: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Property: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Property: \( \log_b(x^a) = a \cdot \log_b(x) \)

These rules apply similarly to natural logarithms, with \( \log_b \) being replaced by \( \ln \). Applying the property \( \ln(e^x) = x \) as seen in our example, \( \ln(e^{-5.03}) \) simplifies directly to \( -5.03 \).
These properties allow us to break down complicated logarithmic expressions into simpler parts. This can be useful not only for simplification but also for solving equations and analyzing growth/decay models.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. An exponent is written as a small number to the upper right of a base number. For example, \( a^n \) means that \( a \) is multiplied by itself \( n \) times. Exponential expressions can grow or shrink very quickly, which is why they are often used in scientific calculations, such as those involving exponential growth or decay.
There are several basic properties of exponents that are helpful to remember:

• \( a^m \cdot a^n = a^{m+n} \)
• \( \frac{a^m}{a^n} = a^{m-n} \)
• \( (a^m)^n = a^{m\cdot n} \)
• \( a^{-n} = \frac{1}{a^n} \) - this property shows how to handle negative exponents

In the expression \( e^{-5.03} \), \( e \) is the base and \( -5.03 \) is the exponent. Since the natural logarithm is the inverse of the exponential function, using \( \ln(e^x) = x \) allows us to simplify \( \ln(e^{-5.03}) \) effectively by recognizing that the exponent indicates an inverse, or reciprocal, relationship here.
Mastering these fundamental properties will greatly enhance your mathematical problem-solving skills, especially in calculus and exponential growth or decay scenarios.