Question

In Exercises, write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 2=0.6931 \ldots $$

Short answer

The exponential form of the given logarithmic equation \(\ln 2 = 0.6931...\) is \(e^{0.6931...} = 2\).

Step-by-step solution

Understand the logarithmic equation

The given equation is in logarithmic form, i.e. \(\ln 2 = 0.6931...\). This means that the base of the logarithm is \(e\), the argument is \(2\), and the logarithm itself is \(0.6931...\). In other words, raising \(e\) to the power of \(0.6931...\) equals \(2\).

Convert logarithmic form to exponential form

The exponential form of a logarithmic equation \(\ln x = y\) is \(e^y = x\). Applying the same rule, the given logarithmic equation \(\ln 2 = 0.6931...\) can be rewritten as \(e^{0.6931...} = 2\).

Key concepts

Natural Logarithm

The natural logarithm is a specific kind of logarithm that is denoted as and has the constant e as its base. It is an invaluable tool in various fields such as mathematics, engineering, and the physical sciences. To understand this concept fully, consider that logarithms, in general, are the inverse operations to exponentiation. This means that while exponentiation involves raising a number to a certain power, logarithms ask the question: 'To what power must we raise the base to obtain a specific number?'

For the natural logarithm specifically, the base is e, which is approximately 2.71828, also known as Euler's number, a fundamental constant in mathematics. When you see an equation such as a = b, what it really says is that e raised to the power b equals a. The natural logarithm of a number x can be represented as x and is the exponent to which e must be raised to produce the number x.

In terms of notation, this can be confusing for students. But remember, when you see without a base, it is implied that the base is e. In the given exercise, the natural logarithm of 2 ( 2) equals approximately 0.6931, indicating that e to the power of 0.6931 equals 2.

Exponential Form Conversion

Converting a logarithmic equation to its exponential form is an essential skill in algebra. It helps you understand the relationship between logarithms and exponents more clearly. When dealing with a natural logarithm, converting to the exponential form involves using Euler's number, e, as the base of the exponent.

Here is the general process for converting a natural logarithmic equation of the form x = y to its exponential form: e raised to the power of y equals x, or e^y = x. This conversion allows us to solve for x when we know y, and vice versa. It's a powerful process because it can also be used when dealing with exponential growth and decay problems, where e often appears due to its natural relationship to these phenomena.

In the exercise provided, 2 = 0.6931 is the logarithmic form. By following the conversion process, we rewrite it as e^{0.6931} = 2. This exponential equation reveals that the variable's value is reached when e is raised to the power of approximately 0.6931.

Euler's Number

Euler's number, denoted as e, is a transcendental number that is the base of the natural logarithm. Its approximate value is 2.71828, and it's one of the most important constants in mathematics, much like pi (). The number e shows up in various areas of mathematics and the natural sciences, prominently featured in calculations involving exponential growth and decay, compound interest, and more.

One reason why e is so special is its rate of growth. When you have an exponential function of the form e^x, the rate at which the function grows is proportional to its current value, which leads to a natural smooth growth pattern. This makes e exceptionally important in calculus, especially when dealing with derivatives and integrals of exponential functions.

In the context of the exercise, understanding Euler's number is critical. It serves as the unwritten base for the natural logarithm in 2 = 0.6931. Recognizing the role of e in converting from logarithmic to exponential form helps you grasp why the exponential equation looks like e^{0.6931} = 2, emphasizing that e raised to the power of 0.6931 equals 2.