Problem 11 - Intermediate Algebra

Question

Fill in the blanks. \(\log _{b} x\) is the _____ to which \(b\) is raised to get \(x.\)

Step-by-Step Solution

Answer

Power

Step 1: Understanding Logarithmic Expression

The expression \( \log_{b} x \) is a logarithmic function where \( b \) is the base, and \( x \) is the number for which we want to find the logarithm. This expression asks the question: 'What power should \( b \) be raised to, in order to result in \( x \)?' This is the fundamental definition of a logarithm.

Step 2: Interpret Logarithm Definition

According to the definition of a logarithm, \( \log_{b} x = y \) means that \( b^y = x \). Thus, \( y \) is the power to which the base \( b \) must be raised to obtain \( x \). The logarithm tells you the exponent.

Step 3: Filling in the Blank

Based on the definition of logarithms, we can conclude that the word that fills the blank in the sentence ' \( \log_{b} x \) is the _____ to which \( b \) is raised to get \( x.\)' is 'power'. This is because the logarithm represents the exponent needed to achieve the number \( x \) when the base \( b \) is raised to it.

Key Concepts

Logarithmic Functions

Logarithmic functions play a fundamental role in mathematics, particularly in areas such as algebra and calculus. When you see a logarithmic expression like \( \log_{b} x \), it is essentially asking the question: 'To what power must the base \( b \) be raised, to result in the number \( x \)?'
This creates a unique relationship between exponentiation and logarithms. Logarithms act as the inverse operation of exponentiation, meaning they undo what exponentiation does.
For example, if \( b^y = x \), then \( \log_{b} x = y \). This equation illustrates how logarithms are used to determine the exponent \( y \) that makes this equality true. The function essentially returns the exponent when provided with a base and the resultant number. Understanding this concept is crucial in the world of higher mathematics.

Base of a Logarithm

The base of a logarithm, denoted by \( b \) in the expression \( \log_{b} x \), is a critical component within logarithmic calculations. It refers to the number that we are raising to a certain power to get another number.
A few points to consider about the base in a logarithm:

• The base \( b \) must be a positive number, typically other than 1.
• Common bases include 10, 2, and the natural base \( e \), where \( e \approx 2.718 \).

The choice of base can drastically affect the outcome of the logarithm. For instance, a base of 10 is often used in scientific notation for simplification and is known as the common logarithm. The natural logarithm, often denoted as \( \ln \), uses the base \( e \) and is crucial in calculus due to its useful derivative and integral properties. By understanding the role and choice of base, one can effectively tackle various logarithmic functions and equations.

Exponentiation

Exponentiation is the process of raising a number, known as the base, to the power of an exponent. This operation forms the core concept behind logarithms. When dealing with numbers, exponentiation allows us to express repeated multiplication simply and efficiently.

• The expression \( b^y = x \) involves exponentiation, where \( b \) is the base and \( y \) is the exponent, resulting in the number \( x \).
• Exponentiation is pivotal because it establishes the relationship that logarithms are built upon, essentially reversing the operation to find the unknown exponent.

When you encounter a logarithmic expression such as \( \log_{b} x \), you're essentially solving for the exponent \( y \) in the context of the equation \( b^y = x \). This interplay between logarithms and exponentiation is what makes these mathematical expressions both powerful and efficient when handling problems that vary exponentially, such as those in science, engineering, and computer science.