Problem 35 - Algebra and Trigonometry

Question

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{5} 5 $$

Step-by-Step Solution

Answer

The value of \( \log _{5} 5 \) equals to 1.

Step 1: Identify the base of the logarithm

The base of the given logarithm is 5 which is denoted by the subscript of the logarithm. Written as \( \log _{5} 5 \).

Step 2: Apply the property of logarithm

Apply the property of logarithms which is \( \log _{b} b = 1 \). In the given problem, \( b \) is 5. So, \( \log _{5} 5 = 1 \).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is essential when you're trying to evaluate expressions like \( \log_{5} 5 \). Logarithms have unique characteristics that simplify complex calculations. One crucial property to remember is the 'identity property' of logarithms, which states that \( \log_b b = 1 \) for any base \( b \). This is because the logarithm answers the question: to what power must we raise the base to get the number itself? Clearly, any base raised to the power of one yields the base; hence, \( b^1 = b \). Another key property is the 'inverse property,' stating that \( b^{\log_b x} = x \), and \( \log_b (b^x) = x \). These properties are particularly helpful in simplifying expressions and solving equations involving logarithms.

Additionally, knowing how to manipulate logarithms using the product, quotient, and power rules can be quite beneficial. For instance, the product rule allows you to transform the logarithm of a product into the sum of logarithms: \( \log_b (MN) = \log_b M + \log_b N \). Similarly, the quotient rule allows you to write the logarithm of a quotient as the difference: \( \log_b (M/N) = \log_b M - \log_b N \). Lastly, the power rule states that the logarithm of a power is equal to the exponent times the logarithm of the base: \( \log_b (M^k) = k \cdot \log_b M \). Memorizing and practicing these rules will make logarithm problems much more manageable.

Logarithm Base

The base of a logarithm is the number that is raised to a power. For instance, in \( \log_{5} 5 \), the base is 5. It's important to note that common logarithms (base 10) and natural logarithms (base \( e \), where \( e \) is Euler's constant) are widely used and often have their own notation: \( \log \) for base 10 and \( \ln \) for base \( e \). Understanding the base is crucial because it determines the value of the logarithm and influences how you apply logarithmic properties and rules.

Different bases can also be converted using the change of base formula: \( \log_b a = \frac{\log_k a}{\log_k b} \), where \( k \) can be any positive number except 1. This is helpful when dealing with bases that aren't directly calculable or when trying to solve an equation that requires both sides to be in the same base. Remember to always match the base with the number inside the logarithm function for direct computation, such as \( \log_{5} 5 \), where it's evident that the base 5 is the foundation of the logarithmic expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that combine numbers, variables, and operators. These expressions can represent relationships between variable quantities and allow for the manipulation of these quantities in equation form. When working with logarithms, they can appear in algebraic expressions, and it's beneficial to understand how to manipulate them according to algebraic laws.

Moreover, algebraic expressions can be expanded or factored through the application of logarithmic properties. For example, log properties can turn a log expression involving a product into an addition of logs, making it easier to handle algebraically. Simplifying algebraic expressions with logarithms might involve combining like terms, using properties of exponents, and utilizing the aforementioned product, quotient, and power rules. This is especially useful for solving equations where a variable is within a logarithm, as handling these expressions correctly can help isolate the variable and find its value. Effective use of algebraic manipulation in logarithmic contexts can significantly ease the process of evaluating expressions and solving complex equations.