Problem 28 - Intermediate Algebra

Question

In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ \log _{7} 1 $$

Step-by-Step Solution

Answer

\( \log_7 1 = 0 \).

Step 1: Identify the logarithmic rule applicable

The expression to evaluate is \( \log_{7} 1 \). According to the logarithmic identity, \( \log_b a = c \) implies \( b^c = a \). One special case of this identity is when the argument of the log is 1. Regardless of the base \( b \), if \( \log_b 1 \) is given, it simplifies to 0.

Step 2: Apply the rule

Using the logarithmic property that \( \log_b 1 = 0 \), where \( b \) is any positive number different from 1, we find that \( \log_7 1 = 0 \). This is because \( 7^0 = 1 \), confirming that the expression evaluates to zero.

Key Concepts

Logarithmic Identity

Logarithmic identities are fundamental rules that help us understand and simplify logarithmic expressions. One important identity is given by the expression \( \log_b a = c \), meaning that the base \( b \) raised to the power \( c \) equals \( a \). For instance, in simpler terms, if you see \( \log_b a = c \), it translates to saying "what power should \( b \) be raised to, in order to get \( a \)?"

A special case of this identity occurs when the logarithm has 1 as its argument, such as \( \log_b 1 \). Using the identity, we conclude that any number \( b \), except 1, raised to the power of 0 equals 1. Consequently, it follows naturally that \( \log_b 1 = 0 \), irrespective of the positive value of \( b \). Understanding this identity simplifies countless mathematical problems and shows the elegant consistency of logarithmic functions across different bases.

Logarithmic Properties

Logarithmic properties are rules that allow us to manipulate and simplify logarithmic expressions effectively. These properties are crucial for solving equations and dealing with complex expressions.

Some key properties include:

• Product Property: \( \log_b (mn) = \log_b m + \log_b n \). This tells us that the logarithm of a product is the sum of the logarithms.
• Quotient Property: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \). Here, the logarithm of a quotient is given by the difference of the logs.
• Power Property: \( \log_b (m^p) = p \cdot \log_b m \). This property allows us to bring down the exponent in a log expression as a multiplier.
• Change of Base Formula: \( \log_b a = \frac{\log_k a}{\log_k b} \). This is useful when calculating logarithms in unfamiliar bases.

By using these properties, we can tackle a broad range of logarithmic problems, transforming them into simpler forms to evaluate or solve. These properties underline the adaptability and powerful nature of logarithms in mathematics.

Base of Logarithm

The base of a logarithm is the fixed number that the logarithm is associated with. In \( \log_b a \), \( b \) is called the base. It's crucial to understand that the base must be a positive number different from 1. This requirement is due to the unique properties of exponential growth and decay.

Common bases you might encounter include:

• Base 10: Known as common logarithms. Denoted as \( \log x \), it's frequently used in science and engineering.
• Base \( e \): Known as natural logarithms, denoted \( \ln x \). The number \( e \) is an irrational constant approximately equal to 2.71828 and is used widely in calculus and mathematical models of continuous growth.
• Base 2: Known as binary logarithms, particularly applicable in computer science and information theory.

The concept of the base is essential because it defines the "world" in which the logarithm is operating. This "world" could be real-life applications, computational systems, or scientific models. Moreover, understanding how the choice of base affects the logarithmic identity \( \log_b a = c \) helps us in manipulating and interpreting data in various disciplines.