Problem 29 - Precalculus

Question

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

Step-by-Step Solution

Answer

The answer is 0.

Step 1: Understand the Logarithm Definition

A logarithm is essentially the exponent to which the base number must be raised to produce a given number. For the expression \( \log_{6}(1) \), we are trying to find the exponent that makes 6 become 1. Mathematically, this is expressed as \( 6^x = 1 \).

Step 2: Recall the Exponent Rule for 1

Any number raised to the power of 0 is equal to 1, i.e., \( a^0 = 1 \) for any non-zero number \( a \). Therefore, \( 6^0 = 1 \).

Step 3: Calculate the Logarithm

From the previous step, since \( 6^0 = 1 \), it follows that the exponent we are looking for is \( 0 \). Thus, \( \log_{6}(1) = 0 \).

Key Concepts

Properties of Logarithms

Logarithms have fascinating properties that make complex calculations more manageable. Understanding these properties is crucial when working with logarithmic expressions. Here are the key properties:\

\
• Product Property: For any positive numbers \(a\), \(b\), and a positive base \(c\) (not equal to 1), the logarithm of a product is the sum of the logarithms: \(\log_{c}(a \cdot b) = \log_{c}(a) + \log_{c}(b)\).
\
• Quotient Property: The logarithm of a quotient is the difference of the logarithms: \(\log_{c}\left(\frac{a}{b}\right) = \log_{c}(a) - \log_{c}(b)\).
\
• Power Property: The logarithm of a power is the exponent multiplied by the logarithm of the base: \(\log_{c}(a^b) = b\cdot\log_{c}(a)\).
\
• Logarithm of One: For any base \(c\), \(\log_{c}(1) = 0\) because any base raised to the power 0 equals 1.
\

\These properties are foundational tools that help transform and simplify logarithmic equations.

Exponent Rules

Exponent rules are essential in understanding how logarithms work since logarithms are inherently linked to exponents. We often apply exponent rules to simplify expressions or when converting between logarithmic and exponential forms. Some fundamental rules include:\

\
• Product of Powers Rule: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
\
• Quotient of Powers Rule: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
\
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \(\left(a^m\right)^n = a^{m \cdot n}\).
\
• Zero Exponent Rule: Any non-zero base raised to the power of 0 is 1: \(a^0 = 1\). This is particularly important when evaluating expressions like \(\log_{6}(1)\), which simplifies directly to 0 because any number to the power of 0 equals 1.
\

These rules make it easier to solve equations and derive solutions, especially when paired with the properties of logarithms.

Logarithm Calculations

Logarithm calculations involve using the properties of logarithms and rules of exponents to derive solutions to problems involving logarithmic expressions. For instance, calculating \(\log_{6}(1)\) demonstrates a direct application of the properties of logarithms and exponent rules.\
\When calculating logarithms:\

\
• Identify the base of the logarithm. This base is the number that must be raised to a certain power to yield the argument of the log function.
\
• Recall that \(\log_{b}(1) = 0\) since any non-zero base raised to the power of 0 equals 1. This simplifies calculations for expressions like \(\log_{6}(1)\).
\
• Apply the zero exponent rule, understanding that \(a^0 = 1\), which confirms the result that \(\log_{6}(1) = 0\).
\

\These steps can aid in solving more complex logarithmic expressions by simplifying them into manageable parts. Understanding the basics of logarithm calculations is essential for tackling advanced math problems efficiently.