Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln e=1$$

Step-by-Step Solution

Answer

The exponential form of the given logarithmic equation \(ln e = 1\) is \(e^{1} = e\).

Step 1: Understand the logarithmic equation

The given logarithmic equation is \(ln e = 1\). Here, 'e' is the base, 1 is the exponent and 'e' is the result of the base 'e' raised to power 1.

Step 2: Write in exponential form

The exponential form of a logarithmic equation is written as: base\(^{exponent}\) = result. So, using the components of our logarithmic equation, we rewrite it as \(e^{1} = e\).

Key Concepts

Logarithms

The concept of logarithms is fundamental to understanding a wide range of mathematical operations. A logarithm answers the question: to what exponent must we raise a given base to obtain a certain number? Expressed in a general form, if we have a logarithm \(\log_b a = c\), it implies that the base \(b\) raised to the power \(c\) equals \(a\) (\((b^c = a)\)).

In the given exercise, \(\ln e = 1\), the 'ln' represents the natural logarithm, which is a logarithm that has the number 'e' as its base. It's important to note that in the case of natural logarithms, the base 'e' is approximately equal to 2.71828, and it's a key constant in mathematics, especially in calculus. The equation tells us that \(e\) raised to the power of 1 yields \(e\), highlighting the identity principle of logarithms where the log of a base to its own value is always 1.

Exponential Equations

Exponential equations involve variables located in the exponent of an expression. They follow the form \(b^x = a\), where \(b\) is the base, \(x\) is the exponent, and \(a\) is the result. Exponential expressions are the inverse operations of logarithms.

To convert from logarithmic form to exponential form, remember the base of the logarithm becomes the base of the exponent. This relationship allows us to switch between expressions depending on the given information or what we are solving for. In our exercise, converting \(\ln e = 1\) to exponential form, \(e^1 = e\), we see the operation directly showcases the nature of the exponential equation; a base (in this case \('e'\)) is raised to an exponent, producing a result.

Natural Logarithm

When dealing with logarithms, the natural logarithm holds a special place and is represented by \(\ln\). This type of logarithm uses the mathematical constant \(e\) as its base. The natural logarithm of a number \(x\) is the exponent to which \(e\) must be raised to yield \(x\), expressed as \(\ln x\).

The equation from our exercise, \(\ln e = 1\), uses the natural logarithm to indicate that \(e\) raised to the power of 1 equals \(e\). This is consistent with the property that the logarithm of 1 to any base is 0 (because any number to the power of 0 is 1), and the logarithm of a base to itself is always 1. It's crucial in calculus and its applications as it is naturally associated with rates of growth and decay, integrals, and many infinite series.