Evaluate each expression log327 log121 log5 1 25 log2128 – As we embark on the evaluation of logarithmic expressions, we enter a realm of mathematical exploration that unveils the intricacies of logarithmic functions and their applications. Logarithms, with their unique properties and ability to simplify complex expressions, provide a powerful tool for understanding the relationships between numbers and their exponents.

In this discourse, we will delve into the evaluation of five specific logarithmic expressions: log327, log121, log5 1, log25, and log2128, illuminating their mathematical significance and practical applications.

Understanding logarithmic expressions is paramount in various scientific and engineering disciplines, enabling researchers and practitioners to analyze data, solve equations, and model real-world phenomena. By mastering the techniques of logarithmic evaluation, we empower ourselves with the ability to unlock the secrets hidden within these expressions and harness their power for problem-solving and knowledge creation.

1. Introduction

The purpose of evaluating the given logarithmic expressions is to understand the concept of logarithms and their properties. Logarithmic expressions are commonly used in various fields such as mathematics, science, and engineering, and it is crucial to comprehend how to evaluate them accurately.

2. Understanding the Expressions

Definition of Logarithms, Evaluate each expression log327 log121 log5 1 25 log2128

Logarithms are mathematical operations that are the inverse of exponentiation. Given a positive number bcalled the base, and a positive number xcalled the argument, the logarithm of xto the base b, denoted as log _bx, is the exponent to which bmust be raised to obtain x.

In other words, log _bx= yif and only if b^y= x.

Base and Argument of the Expressions

The given expressions involve the following bases and arguments:

• log ₃27: base 3, argument 27
• log ₁₂121: base 12, argument 121
• log ₅1: base 5, argument 1
• log ₂128: base 2, argument 128

3. Evaluation Methods

Change of Base Formula

To evaluate logarithmic expressions with different bases, we can use the change of base formula:

log_bx= log _ax/ log _ab

where aand bare positive numbers and a≠ 1.

Steps for Evaluation

To evaluate the given expressions using the change of base formula, we follow these steps:

1. Choose a common base for all the expressions. In this case, we choose base 10 because it is familiar and easy to work with.
2. Use the change of base formula to convert each expression to base 10.
3. Evaluate the resulting expressions using a calculator or logarithmic table.

4. Numerical Evaluation: Evaluate Each Expression Log327 Log121 Log5 1 25 Log2128

Using the change of base formula, we obtain the following numerical values for the given expressions:

• log ₃27 = log ₁₀27 / log ₁₀3 ≈ 3
• log ₁₂121 = log ₁₀121 / log ₁₀12 ≈ 2
• log ₅1 = log ₁₀1 / log ₁₀5 = 0
• log ₂128 = log ₁₀128 / log ₁₀2 ≈ 7

5. Comparative Analysis

From the numerical values, we observe that:

• log ₃27 has the largest value, followed by log ₂128, log ₁₂121, and log ₅1.
• The base of the logarithm affects the value of the expression. As the base increases, the value of the logarithm decreases.
• The argument of the logarithm also affects the value of the expression. As the argument increases, the value of the logarithm increases.

6. Applications

Logarithmic expressions are widely used in various fields, including:

• Mathematics:Logarithms are used in calculus, algebra, and trigonometry to simplify complex expressions and solve equations.
• Science:Logarithms are used in physics, chemistry, and biology to measure quantities that vary over a wide range of values, such as the intensity of sound or the concentration of a chemical solution.
• Engineering:Logarithms are used in electrical engineering, computer science, and other fields to design and analyze systems and algorithms.

Essential Questionnaire

What is the purpose of evaluating logarithmic expressions?

Evaluating logarithmic expressions allows us to determine the exponent to which the base must be raised to obtain the given argument. This process is crucial for simplifying complex expressions, comparing numerical values, and solving equations involving logarithms.

How do we evaluate logarithmic expressions?

Logarithmic expressions can be evaluated using the change of base formula, which involves converting the expression to an equivalent expression with a common base. This formula allows us to utilize the properties of logarithms to simplify the expression and obtain its numerical value.

What are the applications of logarithmic expressions?

Logarithmic expressions find widespread applications in various fields, including mathematics, science, and engineering. They are used in modeling phenomena such as radioactive decay, population growth, and sound intensity. Logarithms also play a crucial role in computer science, cryptography, and financial modeling.