Problem 19 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(\log _{6} 1\)

Step-by-Step Solution

Answer

\(\log_{6} 1 = 0\) because the logarithm of 1 with any positive base is 0.

Step 1: Understand the Properties of Logarithms

Recall that a logarithm \(\log_b a = x\) means \(b^x = a\). Additionally, one fundamental property of logarithms is that \(\log_b 1 = 0\) for any base \(b > 0\). This property arises because any non-zero number raised to the power of 0 is 1. This will help us evaluate the given expression.

Step 2: Apply the Property to the Expression

Given the expression \(\log_{6} 1\), you can apply the property \(\log_b 1 = 0\). Here, the base \(b = 6\) and the expression evaluates to \(0\) because \(6^0 = 1\).

Step 3: Confirm the Result

Check if raising the base, which is 6, to the power of 0 indeed gives 1. Since \(6^0 = 1\), we confirm the result is accurate. Therefore, \(\log_{6} 1 = 0\).

Key Concepts

Logarithmic Expression Evaluation

Evaluating logarithmic expressions can seem tricky at first, but understanding the core concept simplifies the process significantly. To evaluate a logarithmic expression like \(\log_b a\), you're essentially asking, "To what power must the base \(b\) be raised to obtain \(a\)?" Breaking it down further, the logarithm \(\log_b a = x\) implies \(b^x = a\).

Thus, when you encounter \(\log_6 1\), you're looking for the power you raise 6 to, to get 1. Using properties of exponents, the answer becomes evident: any base raised to the power of 0 is always 1. Hence, \(6^0 = 1\), leading us to conclude that \(\log_6 1 = 0\).

So, every time you face a similar expression, remember to reinterpret it: you're simply locating the exponent that transforms the base into the given number, in this case, 1.

Fundamental Properties of Logarithms

Logarithms have some key properties that make them versatile in solving mathematical problems. Understanding and applying these properties can turn complex problems into manageable ones.

Here are some fundamental properties:

• Product: \( \log_b (xy) = \log_b x + \log_b y \)
• Quotient: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \)
• Power: \( \log_b (x^k) = k \cdot \log_b x \)
• Zero: \( \log_b 1 = 0 \), since any base greater than zero raised to the zero power equals one
• Identity: \( \log_b b = 1 \), because raising \(b\) to the first power returns \(b\)

These properties provide the toolkit you need to simplify and evaluate logarithmic expressions more effectively. The zero property, for example, was key in solving \( \log_6 1 \). Acknowledging these fundamental truths will help you build a stronger foundation for understanding and using logarithms in more complex scenarios.

Logarithm Base Properties

Choosing the base of a logarithm can greatly influence the simplicity of evaluating expressions. The base in a logarithmic context is the number that is repeatedly multiplied by itself. Most common bases include 10, known as the "common logarithm", and \(e\) (approximately 2.718), known as the "natural logarithm."

In the context of the expression \(\log_6 1\), the base is 6. The fundamental properties remain consistent, regardless of the base being utilized. The zero property we used, \(\log_b 1 = 0\), states that for any positive base \(b\), raising \(b\) to the zero power will yield 1.

Understanding how different bases can affect logarithmic expressions will allow you to maneuver through various problems with ease. Knowing when to apply bases and how they interact with powers quickly ensures that evaluations become less daunting. It's essential to remember that regardless of whether the base is a standard number like 10, \(e\), or any other positive value, the fundamental characteristics of logarithms hold steady.