Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$e^{1}=e\)

Step-by-Step Solution

Answer

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

Step 1: Identify the base of the exponential function

The base of the exponential function is the value that is raised to some power. Here, the base is 'e'

Step 2: Identify the exponent

The exponent is the value to which the base is raised. In this case, it is '1'

Step 3: Write the equation in logarithmic form

The logarithmic form of an exponential equation \(b^{y}=x\) is written as \(\log _{b} x=y\). So here, the equation becomes \(\log _{e} e = 1\). As we know that logarithm base e is often written as 'ln', the equation is usually written as \(ln (e) = 1\)

Key Concepts

Definition of Logarithm

At its core, a logarithm answers the question: To what exponent do we need to raise a certain base to obtain a given number? The core exercise demonstrates converting an exponential expression into logarithmic form. The definition of a logarithm is built upon this principle.

In mathematical terms, if we have an equation of the form \(b^{y} = x\), where \(b\) is the base, \(y\) is the exponent, and \(x\) is the result, the logarithmic form will be \(\log_b(x) = y\). This translates to 'the logarithm of \(x\) to the base \(b\) equals \(y\).'

For a better understanding, consider an example: to write \(2^3 = 8\) in logarithmic form, it would become \(\log_2(8) = 3\), which means that the power to which base 2 must be raised to result in 8 is 3. Simplifying logarithms involves understanding this conversion from exponential form to logarithmic form, which is crucial in solving equations involving logarithms.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are written in the form \(f(x) = b^x\), where \(b\) is a positive real number not equal to 1, and \(x\) is the exponent. This type of function shows up frequently in various fields including finance, biology, and physics, often to describe growth or decay processes.

In the core exercise example, we examined the base 'e' which is a special mathematical constant. When 'e' is the base, the function \(f(x) = e^x\) is called the natural exponential function, one of the most important functions in mathematics because of its unique properties, especially in calculus. It's the function whose growth rate is proportional to its value, characterizing systems that grow or decay continuously at a constant relative rate.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base 'e', where 'e' is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm of a number \(x\) is the power to which 'e' would have to be raised to equal \(x\). For example, since \(e^1 = e\), the natural logarithm of \(e\) is \(\ln(e) = 1\).

The natural logarithm has various unique properties. It is the inverse function of the natural exponential function, meaning that if \(y = \ln(x)\), then \(e^y = x\). This property allows us to solve exponential equations where the base is 'e' by using natural logarithms. Natural logarithms also have important applications in calculus, such as integration and solving differential equations, because the derivative of \(\ln(x)\) with respect to \(x\) is \(1/x\).