Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$e^{4}=54.5981 \ldots\)

Step-by-Step Solution

Answer

The logarithmic form of the given exponential equation \(e^{4}=54.5981...\) is \(ln(54.5981...) =4\).

Step 1: Identify the Base, Exponent, and Result in the Exponential Equation

In the given equation \(e^{4}=54.5981...\), 'e' is our base, 4 is our exponent, and 54.5981... is the result.

Step 2: Convert the Exponential Equation to Logarithmic Form

Using the formula \(\log_{b}x=y\), substitute the base 'e', exponent 4, and result 54.5981... from the given equation into the formula to obtain the logarithmic equation. So, based on this our equation will become: \(\log_{e}54.5981...=4\).

Step 3: Simplifying the Base 'e' in Logarithmic Form

The base 'e' in logarithmic is commonly known as the natural logarithm and is written as 'ln'. So, we can simplify the equation from \(\log_{e}54.5981...=4\) to \(ln(54.5981...)=4\)

Key Concepts

Understanding Exponential Functions

Exponential functions are fascinating mathematical expressions where a constant base is raised to a variable exponent. This type of function is represented as \(b^x\), where \(b\) is the base and \(x\) is the exponent.

Exponential functions have key characteristics:

• The base \(b\) is a positive real number.
• If \(b > 1\), the function shows exponential growth.
• If \(0 < b < 1\), the function represents exponential decay.

They frequently appear in various fields like biology, finance, and physics due to their nature of modeling growth and decay phenomena.

In our exercise, the exponential function given is \(e^4\), where \(e\) is a special constant approximately equal to 2.71828. This constant is significant in mathematics as it acts as the natural base for exponential functions in calculus.

• Know that understanding the structure of an exponential function is crucial in converting it to logarithmic form later on.
• The base \(e\) plays a vital role in the natural logarithm domain.

Decoding Natural Logarithms

A natural logarithm is a logarithm with the base \(e\), often denoted as \(\ln\).

Natural logarithms are prevalent because they take advantage of the base associated with the natural exponential function. Here are some vital facts about natural logarithms:

• The natural logarithm of a number \(x\) is the power to which the base \(e\) must be raised to yield \(x\).
• Mathematically, it is expressed as \(\ln(x)\).
• When you see \(\ln(x) = y\), it implies \(e^y = x\).

In the context of the exercise, the equation \(e^4 = 54.5981...\) can be represented in natural logarithmic form as \(\ln(54.5981...) = 4\).

Understanding natural logarithms is crucial because they simplify many complex problems involving exponential growth and decay. Additionally, they are used extensively in calculus to describe rates related to growth processes.

The Art of Logarithmic Form Conversion

Logarithmic form conversion is a helpful technique to transform an exponential equation into a logarithmic expression. This conversion is based on the property that if \(b^y = x\), it can be rewritten as \(\log_b(x) = y\).

In the exercise, we see this through these steps:

• Identify the components: base (\