Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{3}=20.0855 . . . $$

Step-by-Step Solution

Answer

The logarithmic form of the given exponential function is \(\ln 20.0855 . . . = 3\).

Step 1: Understanding Exponential and Logarithmic Function

An exponential function is an equation where the base (in this case \(e\)) is raised to a power (in this case 3). A logarithm is a way of expressing the same relationship in a 'flipped' manner.

Step 2: Identify the Base, Exponent and Result

In our given exponential function \(e^3 = 20.0855 . . .\), \(e\) is the base, 3 is the exponent and 20.0855 . . . is the result.

Step 3: Conversion into Logarithmic Form

To convert the exponential equation into logarithmic form, we use the general rule that converts an exponential function \(b^x = y\) to a logarithm: \(\log_{b} y = x\). Applying this rule, we get \(\ln 20.0855 . . . = 3\). \(\ln\) is used for base \(e\), It's the natural logarithm.

Key Concepts

Exponential Functions

Exponential functions are equations where a constant base is raised to a power or exponent. They are widely used to model growth rates in fields such as biology, finance, and physics. In mathematics, the base is often represented by the number 10 or the natural number, e, which is approximately equal to 2.718.

Key characteristics of exponential functions include:

• The base is always a positive constant greater than zero but not equal to one.
• The exponent can be any real number, resulting in varied outputs.
• Exponential functions grow very fast as the exponent increases, especially for larger bases.

Understanding these properties is crucial for transforming them into their equivalent logarithmic form, which serves as the foundation for solving many algebraic problems.

Logarithmic Form

Logarithmic form is a way of expressing the relationship found in an exponential equation in another way. It essentially 'flips' the roles of the exponent and the result. This is accomplished through the logarithm.

For any exponential equation of the form:

• \(b^x = y\)

It can be rewritten in logarithmic form as:

• \(\log_b y = x\)

In this equation, \(\log_b\) denotes a logarithm with base \(b\). This form is particularly useful because it helps isolate the exponent, making it easier to solve for it when given the base and result. Converting exponential equations into logarithmic form provides a simpler method for calculating unknown quantities.

Natural Logarithm

The natural logarithm is a special type of logarithm that uses the mathematical constant \(e\) as its base. The natural logarithm is denoted by \(\ln\) and is the inverse operation of the exponential function using base \(e\).

Here's why natural logarithms are important:

• They commonly occur in mathematical modeling for phenomena that continuously grow or decay, such as populations, radioactive decay, and even financial interest.
• The base \(e\), is highly significant in calculus and complex mathematical functions for its unique properties.

When converting an expression like \(e^x = y\) to logarithmic form, it translates to \(\ln y = x\). Recognizing and using natural logarithms allows us to approach complex exponential relationships effectively.

Base and Exponent Conversion

Converting between base and exponent forms, particularly between exponential and logarithmic forms, is critical in understanding exponential equations. This conversion allows us to unravel the components of an equation by translating them into a more manageable form.

Understand that in any exponential equation:

• \(b^x = y\)

The conversion into a logarithmic equation is done using the formula:

• \(\log_b y = x\),

which projects the exponent as the result of a logarithmic operation. The ability to easily switch between these forms can simplify the process of solving equations and provides a deeper comprehension of the underlying relationships in mathematics. It makes solving for unknown variables in complex equations more straightforward and efficient.