Problem 12 - Precalculus

Question

Write each logarithmic equation in its equivalent exponential form. $$1=\ln e$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Logarithmic Equation

We start with the equation given: \(1 = \ln e\). The natural logarithm, \(\ln\), is the logarithm with base \(e\). The equation \(1 = \ln e\) represents a relationship where the natural logarithm of a number is equal to 1.

Step 2: Identifying the Base of the Logarithm

Since \(\ln\) denotes the natural logarithm, the base is always \(e\). Thus, our equation \(1 = \ln e\) can be expressed with base \(e\) as the log base: \( \ln e = 1\).

Key Concepts

Exponential Form

Exponential form allows us to express a logarithmic equation in terms that highlight the relationship between the base of the logarithm, the exponent, and the result. This is particularly helpful for solving or simplifying equations. When a logarithmic equation like \(1 = \ln e\) is given, to convert it into exponential form, we relate the concepts:

• The base is \(e\), since it is a natural logarithm.
• The result of the logarithm is 1.
• The corresponding exponential form is \(e^1\).

Thus, the equivalent exponential form of \(1 = \ln e\) is \(e^1 = e\). This conversion reveals the direct correlation between the components of the logarithm's base, the exponent, and the output value.

Natural Logarithm

A natural logarithm is a logarithm to the base \(e\), which is approximately equal to 2.71828. It is denoted by \(\ln\). Natural logarithms are common in mathematics, particularly in calculus, due to their unique properties that make calculations involving exponential growth and decay simpler. For any number \(x\), \(\ln x\) represents the power to which \(e\) must be raised to yield \(x\).In the case of our example, \(\ln e\) asks us what power \(e\) must be raised to, in order to equal \(e\) itself. The answer is 1, because raising any number to the power of 1 yields that number itself. Hence, \(\ln e = 1\) essentially states that the logarithm of \(e\) with base \(e\) results in 1.

Base e

The base \(e\) is a fundamental constant in mathematics, often referred to as Euler's number, named after the Swiss mathematician Leonhard Euler. Its approximate value is 2.71828, and it is an irrational number, which means it cannot be written as a simple fraction.Euler's number \(e\) is significant in many areas of mathematics:

• It appears in calculations of compound interest, where continuous compounding grows at a rate \(e\).
• It is prevalent in calculus, especially in functions like \(e^x\) and their derivatives and integrals.
• It underpins the natural logarithmic function, \(\ln x\), making it essential for solving growth and decay problems.

The beauty of \(e\) lies in its natural occurrence across diverse mathematical disciplines, aiding in understanding phenomena that exhibit exponential behavior, such as population growth and radioactive decay.