Problem 14 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\ln (e)=1\)

Step-by-Step Solution

Answer

The equation \( \ln(e) = 1 \) becomes \( e^1 = e \).

Step 1: Understanding the Problem

The given equation is \( \ln(e) = 1 \). This is a logarithmic equation. Our task is to rewrite it in the form of an exponential equation using the theorem \( b^a = c \) if and only if \( \log_b(c) = a \).

Step 2: Identifying Components

The natural logarithm \( \ln(x) \) is the same as \( \log_e(x) \). Thus, \( \ln(e) = 1 \) can be rewritten as \( \log_e(e) = 1 \). Here, \( b = e \), \( c = e \), and \( a = 1 \).

Step 3: Rewrite as an Exponential Equation

Using the property \( \log_b(c) = a \) if and only if \( b^a = c \), we substitute \( b = e \), \( a = 1 \), and \( c = e \) to get \( e^1 = e \).

Step 4: Verify the Equation

To ensure the conversion is correct, we see that \( e^1 = e \) is a true statement, confirming the correct conversion of logarithmic to exponential form.

Key Concepts

Logarithmic Equations

Logarithmic equations involve expressions that contain logarithms, such as \( \ln(x) = y \) or \( \log_b(x) = y \). In these equations, the goal is often to find the unknown value, typically represented by \( x \), given a specific base and result. In the initial exercise, we have the equation \( \ln(e) = 1 \), a classic example of a logarithmic equation. This tells us that when the base \( e \) is raised to the power that gives \( 1 \), we find the natural log of \( e \) is 1. Understanding the structure of logarithmic equations requires recognizing that:

• \( \log_b(x) \) represents the exponent you put on \( b \) to get \( x \).
• \( \ln(x) \) is a specific logarithmic equation where the base is the natural number e, approximately 2.718.

In practice, logarithmic equations allow us to solve for unknown exponents in exponential scenarios by providing a method to reframe the equation to be more manageable.

Conversion Between Forms

Converting between exponential and logarithmic forms is a powerful algebraic tool. This skill helps in simplifying and solving equations that might otherwise be complex. The theorem \( b^a = c \) if and only if \( \log_b(c) = a \) provides a simple pivot point:

• The exponential form \( b^a = c \) asks 'what is \( c \) when \( b \) is raised to \( a \)?'
• The logarithmic form \( \log_b(c) = a \) asks 'what power do we raise \( b \) to get \( c \)?'

In the given exercise context, the conversion \( \ln(e) = 1 \) from logarithmic to exponential form is straightforward: knowing that the natural log's base is \( e \), seeing \( \ln(e) = 1 \) converts to \( e^1 = e \). Remember:- Rewriting forms can help significantly in solving problems.- It's like translating mathematical languages, each suited to different types of problems.

Natural Logarithm

The natural logarithm, denoted by \( \ln(x) \), is a logarithm with base \( e \), where \( e \approx 2.718 \), a mathematical constant also known as Euler's number. This form of logarithm appears often in natural growth processes, finance, and other exponential functions. Key aspects of the natural logarithm include:

• \( \ln(e) = 1 \): Any number raised to the first power is itself, so \( e^1 = e \).
• This property illustrates the direct relationship between exponential and logarithmic forms, simplifying conversions.
• Natural logs are used to solve for time in compound interest or decay processes, as they reverse exponential growth or decay.

Understanding the natural logarithm requires familiarity with exponential growth, which appears naturally in many scientific and financial contexts. Recognizing when to use \( \ln(x) \) helps in solving real-world problems effectively.