Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{5} 125\)

Step-by-Step Solution

Answer

The value of \(\log _{5} 125\) is 3.

Step 1: Initial Understanding

The logarithmic expression \(\log _{5} 125\) means 'to what power should we raise 5 to get 125'

Step 2: Solve for the power

By observation or knowing the cube of 5, it is clear that \(5^3 = 125\)

Step 3: Apply logarithmic identity

Applying the basic logarithmic identity, if \( b^y = x \) then \( \log _{b} x = y \), where b is the base, x is the argument and y is the exponent. For our case, \(log _{5} 125 = 3\) since \(5^3 = 125\)

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, specifically within the field of algebra. They serve as an inverse operation to exponentiation. This means that where exponentiation involves raising a number (known as the base) to a power to get another number, logarithms help us find that power when the base and the result are known.

Let's simplify this with an example. In our exercise, the logarithmic expression \(\log _{5} 125\) asks for the power to which the base, 5, must be raised to produce the number 125. So, when you see the expression \(\log _{5} 125\), think of it as a question: 'What exponent do we apply to the number 5 to get 125?' The answer to this question, as we explore in the following sections, simplifies the process of solving logarithmic expressions.

Exponents

Exponents are a succinct way to express repeated multiplication of a number by itself. When we talk about 5 raised to the power of 3, denoted as \(5^3\), we mean 5 multiplied by itself two more times, which is \(5 \times 5 \times 5 = 125\).

Understanding exponents is crucial because they are intrinsically linked to logarithms. Knowing the relationship between a base and its exponent helps us not only solve for the missing exponent given a base and its result but also allows us to simplify expressions and solve equations that might seem complex at first glance. The ability to mentally or quickly calculate smaller exponents can greatly assist in understanding more advanced mathematical concepts.

Logarithmic Identity

Logarithmic identities are mathematical statements that equate two expressions and provide a clearer view of the properties and operations of logarithms. An essential logarithmic identity that we use in the exercise is: if \( b^y = x \), then \( \log _{b} x = y \). This identity is the bedrock for understanding how to evaluate logarithmic expressions.

The reason logarithmic identities are so critical is that they bring out the true essence of logarithms: uncovering the exponents. The equation provided by the identity unfolds the definition of the logarithm itself and helps students grasp why \( \log _{5} 125 = 3 \) is valid; it's because \(5^3 = 125\). This particular logarithmic identity serves as a bridge between the concepts of exponents and logarithms and is a potent tool in the simplification and solution of logarithmic problems.