Problem 38 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{6} 1 $$

Step-by-Step Solution

Answer

The value of the expression \( \log _{6} 1 \) is 0.

Step 1: Identify the Expression

We need to evaluate or simplify: \(\log _{6} 1\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

The value of the expression \( \log _{6} 1 \) is 0.

Key Concepts

Evaluating Logarithmic Expressions

Understanding how to evaluate logarithmic expressions is crucial for solving many problems in mathematics. A logarithmic expression represents the exponent to which a base number is raised to get a certain value. Let's simplify this concept with an example.

Consider the logarithm expression \( \log_{6}1 \). To evaluate this without a calculator, we refer to one of the fundamental properties of logarithms: any base raised to the power of 0 equals 1. It's like asking, 'What power should we raise 6 to, to get 1?' The answer is straightforward: 0, because \( 6^0 = 1\). Thus, \( \log_{6}1 = 0\). When you face a logarithmic expression, always remember that the logarithm of 1 to any base is zero. This understanding can simplify more complex problems and is a stepping stone to mastering logarithms.

Logarithm Rules

The rules of logarithms, also known as logarithm laws or identities, are essential for simplifying and solving logarithmic expressions effectively. Here are some key rules to understand:

• The Product Rule: \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \), dictating that the logarithm of a product is the sum of the logarithms.
• The Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), stating that the logarithm of a quotient is the difference between the logarithms.
• The Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \), which tells us that the logarithm of a power is the exponent times the logarithm of the base.

These rules are pillars for working with logarithmic expressions. Whenever you encounter complex logarithms, try to simplify them using these three powerful tools. Remember, understanding these rules is like having a math toolkit that can dismantle many complicated problems into manageable parts.

Base of Logarithms

The base of a logarithm is the number that is raised to a certain power to obtain the specific value in the expression. For instance, in \( \log_{6}1 \) from the original exercise, 6 is the base. The choice of base greatly affects the value of the logarithm.

A common base is 10, also known as the common logarithm, often written as \(\log\) without a base specified. Another one is the natural logarithm with base \(e\), an irrational constant approximately equal to 2.718, denoted by \(\ln\).

Different bases can be interconverted using the change of base formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \(b\) and \(c\) are different bases. It's vital to understand the concept of the base as it represents the foundation upon which the logarithmic structure is built. To grasp logarithms thoroughly, start by practicing with different bases to witness how they alter the outcomes of the expressions you are working with.