Problem 29 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{10} 1 $$

Step-by-Step Solution

Answer

The value is 0.

Step 1: Understand the Logarithm

A logarithm \(\log_{b}(a)\) is the exponent to which the base \( b \) must be raised to yield \( a \). Therefore, the expression \(\log_{10} 1\) means finding the exponent \( x \) such that \( 10^x = 1 \).

Step 2: Base Case of Logarithm

Recall the base case for any logarithm: the logarithm of 1 with any base is 0. This is because any number raised to the power of 0 is 1.

Step 3: Application to the Problem

Apply the base case to the current problem: since \( 10^0 = 1 \), it follows that \(\log_{10} 1 = 0 \).

Key Concepts

Base of a Logarithm

In the context of logarithms, the term 'base' is crucial. It refers to the number that needs to be raised to a certain power to result in another number. When you see a logarithm expressed as \( \log_b(a) \), the \( b \) is called the base. Here, the goal is to find an exponent that makes \( b \) reach the value of \( a \).
This base is foundational to understanding how logarithms function. For instance, in \( \log_{10} 100 \), 10 is the base. It’s asking, “To what power should 10 be raised to get 100?” Since \( 10^2 = 100 \), the answer is 2.
It's important to keep in mind:

• A logarithm can be considered the opposite of exponentiation. It asks the question of 'how many times'
• Bases are always positive numbers and not equal to one

Understanding this concept helps in solving logarithmic expressions effectively.

Exponentiation

Exponentiation is the process of raising a number to a power. In simpler terms, it's a way to express repeated multiplication. For instance, \( 2^3 = 2 \times 2 \times 2 = 8 \).
This concept is fundamental when dealing with logarithms because a logarithm asks you to find the exponent. The relationship can be seen as:

• If \( b^x = a \), then \( \log_b(a) = x \).
• This equation highlights that the logarithm gives us the value of the exponent \( x \).

Consider how powerful exponentiation is in reducing complex multiplication into a more manageable format. It simplifies processes in both mathematics and real-world applications like calculating compound interest.
When tackling problems involving logarithms, understanding exponentiation can provide clarity. It serves as a bridge between the question asked by the logarithm and the numerical answer derived from solving it.

Base Case for Logarithms

The base case for logarithms is a fundamental principle that makes many logarithmic calculations straightforward. Essentially, when you encounter \( \log_b(1) \), the answer is always 0, because any base raised to the zero power is 1.
This case simplifies the work:

• It's a universal rule applicable for any base \( b \) — whether it be 2, 10, or 100.
• This rule comes in handy, saving time when evaluating logarithmic expressions.

For example, in the problem \( \log_{10} 1 \): you’re asked to find the power to which 10 must be raised to yield 1. By the base case principle, the answer is 0, since \( 10^0 = 1 \).
Recognizing and applying base case scenarios helps efficiently solve logarithms without needing complex calculations every time. It reinforces an understanding of how logarithms and exponentiation interrelate.