Problem 56 - College Algebra

Question

Let \(b\) be any positive real number such that \(b \neq 1\). What must \(\log _{b} 1\) be equal to? Verify the result.

Step-by-Step Solution

Answer

\(\log_b 1 = 0\). It verifies since \(b^0 = 1\).

Step 1: Understanding the Logarithmic Expression

We need to find \(\log_b 1\), which is asking the question, 'To what power must \(b\) be raised to get 1?' A logarithm essentially reverses the process of exponentiation.

Step 2: Applying the Logarithmic Identity

Recall the fundamental property of logarithms that states \(\log_b 1 = 0\) for any base \(b > 0, b eq 1\). This is because any number raised to the power of zero equals one, i.e., \(b^0 = 1\).

Step 3: Verification of the Result

To verify, let's calculate with base exponentiation: For any \(b > 0\), \(b^0 = 1\). Thus, \(\log_b 1\) indeed equals 0, confirming that a correct application of the properties of logarithms gives us the expected result.

Key Concepts

Exponentiation

Exponentiation is a mathematical process involving raising a number, known as the base, to a specific power, referred to as the exponent. When we talk about exponentiation, we are essentially discussing repeated multiplication.
For example, if we have a base of 2 and an exponent of 3, we calculate it as:

• \(2^3 = 2 \times 2 \times 2 = 8\)

Exponents are powerful because they allow us to express large numbers concisely. A key thing to remember is that any number raised to the power of zero is always 1. This is crucial for understanding logarithms, as it is a foundational concept within the process of reversing exponentiation. For example:

• \(b^0 = 1\)

This property helps pave the way for understanding why \(\log_b 1 = 0\) across all bases other than 1.

Logarithmic Expressions

Logarithmic expressions provide a way to solve for exponents within mathematical equations. When we encounter a logarithm, it is asking us, "To what exponent should we raise the given base to obtain a certain number?"
The expression \(\log_b x\) stands for the power or exponent to which the base \(b\) must be raised to yield \(x\). For instance, if \(\log_2 8 = 3\), it means 2 must be raised to the power of 3 to obtain 8.

• Key Concept: \(b^y = x\) can be rewritten as \(\log_b x = y\).
• This is the reverse operation of exponentiation.

To solve \(\log_b 1\), you're essentially determining the exponent applied to base \(b\) to result in 1. Recognizing this helps you connect logarithms closely with the concept of exponentiation, as they are two sides of the same coin.

Fundamental Property of Logarithms

The fundamental property of logarithms is essential for simplifying and solving logarithmic expressions. One key property is that the logarithm of 1, no matter the base \(b\), is always 0, as long as the base is positive and not equal to 1. This is expressed as:

• \(\log_b 1 = 0\)

This property arises because any number raised to the power of zero equals one, which aligns with exponentiation principles:

• \(b^0 = 1\)

This concept is fundamental in mathematics because it locks the behavior of logarithms into something predictable and reliable, ensuring consistency across various mathematical problems. Understanding these properties allows you to manipulate and solve equations involving logarithms more effectively, confirming that logarithms truly are the natural inverse of exponentiation.