Problem 18 - A Graphical Approach to College Algebra

Question

For each statement, write an equivalent statement in exponential form. $$\ln e^{6}=6$$

Step-by-Step Solution

Answer

\( e^{6} = e^{6} \) in exponential form.

Step 1: Understand the Natural Logarithm

The statement given is \( \ln e^{6} = 6 \). The natural logarithm, \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). The natural logarithm of a number is the power to which \( e \) must be raised to produce that number.

Step 2: Identify the Equivalent Exponential Form

We start from the statement \( \ln e^{6} = 6 \). By definition of logarithms, the statement \( \ln x = y \) is equivalent to saying \( e^{y} = x \). So, \( \ln e^{6} = 6 \) implies that \( e^{6} = e^{6} \), which is a tautological statement and therefore equivalent.

Step 3: Rewrite in Exponential Form

Based on Step 2, the equation \( \ln e^{6} = 6 \) is equivalent to saying \( e^{6} = e^{6} \). Thus, we can rewrite the statement directly as \( e^{6} = e^{6} \) in exponential form.

Key Concepts

Natural Logarithm

Natural logarithms are a fundamental part of mathematics and are commonly represented with the symbol \( \ln \). They have a special base, denoted by the letter \( e \), which is an irrational number approximately equal to 2.71828. This base \( e \) is unique because it is closely related to natural growth processes, such as compound interest and population growth.

The statement \( \ln e^{6} = 6 \) means that the base \( e \) raised to the power of 6 is equal to \( e^{6} \). The expression highlights the logarithm's role in finding how many times \( e \) should be multiplied by itself to reach a particular number. In this case, when \( e \) is raised to the 6th power, it results in \( e^{6} \).

Natural logarithms simplify calculations involving exponentials. For instance, they are heavily used in calculus and complex mathematical modeling. By understanding how they work, you can easily switch from logarithmic form to exponential form and vice versa.

Equivalent Statements

In mathematics, equivalent statements are those that communicate the same mathematical fact, even though they might look different at first glance.

Understanding equivalency is crucial because it allows us to transform one mathematical statement into another, helping solve problems more easily or express ideas more clearly. For example:

• \( \ln x = y \) is equivalent to \( e^{y} = x \). This transformation is essential for solving equations involving logarithms and exponentials.
• In the exercise, \( \ln e^{6} = 6 \) is equivalently written as \( e^{6} = e^{6} \), demonstrating a straightforward conversion between forms.

This practice of finding and using equivalent statements can significantly aid in understanding mathematics. It allows students to perceive different forms of a solution and apply the most suitable one to their specific problem.

Logarithms Definition

Logarithms are the inverse operations of exponentiation. Put simply, if you have an expression of the form \( b^y = x \), the logarithm helps you find the value of \( y \) for given numbers \( b \) and \( x \).

Logarithms answer the question: "To what power should the base \( b \) be raised to achieve the number \( x \)?" This makes them extremely powerful in solving equations where such values are unknown.

• For example, \( \log_{b} x = y \) means \( b^y = x \).
• For natural logarithms, the base is \( e \), so \( \ln x = y \) implies \( e^y = x \).

Understanding the definition and operation of logarithms is crucial to solving many real-world problems, particularly those involving exponential growth or decay.

This exercise demonstrates defining a natural logarithm and rewriting it in its exponential form, helping reinforce the understanding and practical usage of logarithms.