Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\ln \frac{1}{e}=-1\)

Step-by-Step Solution

Answer

The exponential form of the logarithmic equation \(\ln \frac{1}{e} = -1\) is \(e^{-1}=\frac{1}{e}\).

Step 1: Understand the Base

First, recognize that the logarithm in the given equation is a natural logarithm, denoted by \(\ln\). The base of a natural logarithm is the mathematical constant \(e\), approximately equal to 2.71828. So, we rewrite the log equation with the base recognized, \(\log_{e} \frac{1}{e} = -1\).

Step 2: Convert to Exponential Form

According to the definition of a logarithm, \(\log_{e} \frac{1}{e} = -1\) can be rewritten in exponential form as \(e^{-1}=\frac{1}{e}\). This is done by taking the base (e), raised to the power of the right side of the equation (-1), to get the number inside the log on the left side (\(\frac{1}{e}\))

Step 3: Validate the Answer

To confirm the result, you can use the property \(e^{-1} = \frac{1}{e}\), which means that \(e^{-1}\) and \(\frac{1}{e}\) are indeed equal. Thus, our conversion of the original log equation to exponential form is correct.

Key Concepts

Exponential Functions

Exponential functions are a fascinating area of mathematics, and they can be thought of as the opposite of logarithmic functions. In an exponential function, the independent variable is an exponent. Formally, an exponential function has the form \(f(x) = a \cdot b^{x}\), where \(a\) is a constant, \(b\) is the base (a positive real number not equal to 1), and \(x\) is the variable exponent. These functions model growth or decay processes such as population growth, radioactive decay, and interest calculations.

Key features of exponential functions include:

• **Rapid Growth/Decay:** They can grow or decrease exponentially, which means very quickly.
• **Horizontal Asymptote:** As \(x\) approaches infinity or negative infinity, the graph of the function approaches a horizontal line.
• **Base:** The base of the exponential function determines if it represents growth (\(b > 1\)) or decay (\(0 < b < 1\)).

Understanding exponential functions is crucial since they often appear in various scientific and real-world scenarios, providing insight into processes that change rapidly over time.

Natural Logarithm

The natural logarithm is a specific type of logarithm where the base is Euler's number \(e\), which is approximately equal to 2.71828. It's usually denoted as \(\ln\), hence when you see an equation like \(\ln x\), it means \(\log_{e}x\).

The natural logarithm has several important properties:

• **Simplifies Exponential Equations:** It helps in solving equations involving exponential expressions, as it is the inverse operation to raising \(e\) to a power.
• **Use in Calculations:** It is prevalent in calculus, particularly for dealing with problems involving growth rates, such as compounding interest or natural growth processes.
• **Relationship with Other Logs:** The natural logarithm can be expressed in terms of other logarithms. For example, \(\ln x = \log_{10} x / \log_{10} e\).

Natural logarithms are especially useful because they relate directly to exponential growth, and they simplify mathematical models of growth, making complex computation more accessible.

Mathematical Constant e

The mathematical constant \(e\) is one of the most significant numbers in mathematics. Like \(\pi\), \(e\) is an irrational number, which means its exact form cannot be expressed as a fraction. The value of \(e\) is approximately 2.71828.

It is mainly used in natural logarithms and as a base for exponential functions.

• **Properties of \(e\):**
  
  • **Irrationality:** Cannot be precisely written as a fraction.
  • **Exponent for Natural Growth:** Provides a framework for modeling continuous growth. For instance, the function \(e^{x}\) describes situations of continuous growth or decay.
• **Applications:**
  
  • **In Calculus:** Essential for defining the natural exponential function and for calculus operations involving growth and decay processes.
  • **Financial Models:** Frequently used in compounding interest calculations.

The constant \(e\) plays a critical role in mathematics, offering a foundation for various formulas and concepts that describe real-world phenomena.