Problem 21 - 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 \(5^{3}=125\).\(\ln e=1\)

Step-by-Step Solution

Answer

The exponential form of \(\ln e = 1\) is \(e^1 = e\).

Step 1: Identify the base, exponent and the result

Here, we are given \(\ln e = 1\). The natural logarithm \(\ln\) is logarithm in base \(e\). So, the base is \(e\), the result is \(e\) again and the exponent is 1.

Step 2: Apply the logarithm definition

To express this in exponential form, let's apply the logarithm definition which states 'if \(\log_b a = c\), then \(b^c = a\)' where \(b\) is the base, \(c\) is the exponent and \(a\) is the result. In this case, the base \(b\) is \(e\), the exponent \(c\) is 1 and the result \(a\) is \(e\). So applying the logarithm definition, the solution will be \(e^1 = e\).

Key Concepts

Exponential Form

Exponential form is a way of expressing mathematical expressions using a base and an exponent. This form is also known as "power form" as it involves raising a base number to a certain power. For example, the expression \(a^b\) can be defined as "\(a\) raised to the power of \(b\)". This format provides a compact way to represent repeated multiplication of a number by itself.

In the context of logarithms, expressing a logarithmic equation in exponential form allows us to understand the relationship between the base, the exponent, and the result in a more intuitive way. For instance, if we start with a logarithmic equation \(\log_b a = c\), converting it to exponential form gives us \(b^c = a\).

Let's consider the equation from the original problem \(\ln e = 1\):

• "ln" denotes the natural logarithm, which is logarithmic with base \(e\).
• Therefore, when converted to the exponential form, the expression becomes \(e^1 = e\).

This conversion is pivotal for solving many types of problems involving logarithms and helps to simplify and solve equations by rewriting them in an alternate but equivalent form.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special type of logarithm where the base is the mathematical constant \(e\). This constant \(e\) is approximately equal to 2.71828 and is an irrational number made famous due to its deep connections with growth processes, such as in natural and exponential growth situations.

Natural logarithms are used frequently in calculus and complex analysis due to their special properties and ease of differentiation. The function \(\ln x\) is the inverse of the exponential function \(e^x\).

• The natural log of 1 is always 0, i.e., \(\ln 1 = 0\), since \(e^0 = 1\).
• The natural log of \(e\) is always 1, i.e., \(\ln e = 1\), since \(e^1 = e\).

These properties make the natural logarithm extremely useful for converting multiplicative relationships into additive ones, which simplifies the manipulation and solving of equations involving exponentials.

Logarithm Definition

A logarithm is essentially the opposite of exponentiation, and it answers the question: "To what power must the base be raised, to produce a given number?" The general form of a logarithm is \(\log_b a = c\), which translates to "\(b\) raised to the power of \(c\) equals \(a\)" or \(b^c = a\).

The definition helps in converting from a logarithmic form to an exponential form and vice versa. For many students, understanding this conversion is key to mastering the topic of logarithms. It is useful in solving equations where unknowns appear in exponents.

• The base \(b\) is the number which is raised to the power.
• The exponent \(c\) is the power to which the base \(b\) is raised.
• The result \(a\) is the outcome of the base raised to the exponent.

For example, with \(\ln e = 1\), applying the logarithm definition gives us that \(e\) (base) raised to the power of 1 (exponent) results in \(e\). This illustrates how logarithms describe the process of getting to a number by raising another number to a power, encapsulating the idea of repeated multiplication in a form that emphasizes the exponentiation process.