Problem 22 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{5} \frac{1}{125}$$

Step-by-Step Solution

Answer

-3

Step 1: Write the fraction in the logarithm as a power of 5

The number 125 is \(5^3\), hence \( \frac{1}{125} = \frac{1}{5^3} = 5^{-3}\).

Step 2: Apply the logarithm to the base exponent

Using the property of logarithms \( \log_b(b^a) = a \), our equation becomes \( \log_5(5^{-3}) = -3 \)

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is critical for solving many mathematical problems efficiently. A logarithm tells us the power to which a number (the base) must be raised to produce another number. It's essentially asking the question: 'To what exponent should I raise the base to get this other number?'

For example, in the expression \( \log_b{a} \), \(b\) is the base, and \(a\) is the number we're trying to match by raising \(b\) to a certain power. This expression is equal to the exponent \(x\) where \(b^x = a\). So if we look at \(\log_5{\frac{1}{125}}\), we're trying to find the exponent that 5 is raised to in order to get \(\frac{1}{125}\). With a solid understanding of what logarithms represent, we can start exploring how their properties are used to simplify and evaluate them.

Exponent Properties

Exponents play a central role in mathematics, especially when working with logarithms. Some key properties we need to understand include the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\), and the power rule, which allows us to take an exponent to another power by multiplying the exponents together, as in \( (a^m)^n = a^{m \times n} \).

These properties help simplify complex expressions, making them easier to manage and solve. For instance, recognizing that \(\frac{1}{125}\) is the same as \(5^{-3}\) because \(125 = 5^3\), and using the rule for negative exponents, allows us to reframe the problem in terms of a base and its exponent, which is a straightforward path to finding the value of a logarithmic expression.

Logarithms Without a Calculator

Learning to evaluate logarithms without a calculator is an important skill, as it reinforces the understanding of logarithmic relationships and exponent rules. When a calculator is not at hand, or when the problem structure is simple and follows certain patterns, you can find the exact value by wrapping your head around these concepts.

To solve \(\log_5{5^{-3}}\) without a calculator, relate the base of the logarithm (5) to the number you're taking the log of \(\left(5^{-3}\right)\). Here, since the base and the number have the same base (5), the logarithm simplifies to the exponent \(\left(-3\right)\) alone. This method relies on recognizing patterns and using the definition of logarithms effectively.

Base Exponent in Logarithms

The relationship between the base and the exponent in logarithms is the linchpin in solving them. When a logarithmic expression has a base \(b\) and an argument \(b^{a}\), as per the logarithmic identity \(\log_b{b^{a}} = a\), the answer simplifies directly to the exponent \(a\).

This identity lies at the heart of simplifying logarithmic expressions and is exemplified in our problem \(\log_5{5^{-3}} = -3\). Recognizing when the base of the logarithm and the base of the exponent match allows immediate simplification, turning what might seem like a daunting logarithmic problem into a straightforward evaluative step.