Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in exponential form. For example, the exponential form of \(\log _{5} 125=3\) is \(5^{3}=125\).\(\log _{10} 1000=3\)

Step-by-Step Solution

Answer

The exponential form of \(\log _{10} 1000=3\) is \(10^{3}=1000\).

Step 1: Identify the base, exponent, and result

In the logarithmic equation \(\log _{10} 1000=3\), 10 is the base, 3 is the exponent, and 1000 is the result.

Step 2: Convert to exponential form

The exponential form of the logarithm is derived by putting the base raised to the power of exponent and equating it to the result. So the exponential form of \(\log _{10} 1000=3\) is \(10^{3}=1000\)

Key Concepts

Exponential Form

An exponential form represents a number as the result of a base raised to a power, known as the exponent. It’s a concise way of expressing repeated multiplication of the same factor. For example, the expression \(5^3 = 125\) is in exponential form. This tells us that the base number 5 is multiplied by itself three times, resulting in 125.When converting a logarithmic equation to exponential form, you are essentially rewriting the equation to highlight the relationship between these numbers:

• Base: The number being multiplied.
• Exponent: The number indicating how many times the base is multiplied by itself.
• Result or Power: The final value obtained after the multiplication.

Understanding exponential forms is essential for working with logarithms, as it allows you to easily switch between these two types of expressions.

Base of Logarithm

The base of a logarithm is a fundamental component in both logarithmic and exponential expressions. It serves as the backbone of the logarithm, indicating the number that's repeatedly multiplied. In the equation \(\log_{10} 1000 = 3\), the base is 10.The concept of a base is crucial because it sets the scale for the logarithm. Different bases can completely change the interpretation of a logarithmic expression:

• A base of 10 is commonly used and is called the common logarithm.
• The natural logarithm has a base of \(e\), approximately 2.718, and appears frequently in mathematical and scientific applications.
• Other bases, such as 2, can appear in computer science and binary systems.

Grasping the idea of the base of a logarithm helps you understand its properties and how it forms connections with exponential forms.

Logarithmic Equation

A logarithmic equation expresses the idea of finding the exponent that transforms the base into the resultant number. It is often written in the form \(\log_b(a) = c\), where \(b\) is the base, \(a\) is the result, and \(c\) is the exponent. For instance, \(\log_{10} 1000 = 3\) indicates that raising 10 to the power of 3 equals 1000.Converting a logarithmic equation to exponential form involves rewriting it so that the same relationship between \(b\), \(a\), and \(c\) is expressed as an exponential equation:

• You begin with the base.
• Raise it to the exponent.
• Set it equal to the result.

Understanding logarithmic equations is pivotal for solving logarithmic problems because it provides a different perspective on how large or small numbers relate to one another. These equations are the foundation for many scientific, engineering, and mathematical applications.