Problem 36 - Algebra for College Students

Question

Evaluate each logarithmic expression. $$ \log _{5}\left(\frac{1}{25}\right) $$

Step-by-Step Solution

Answer

-2

Step 1: Understand the Logarithm Definition

The logarithmic expression \( \log_{b}(x) \) asks, "To what power should the base \( b \) be raised to yield \( x \)?" Here, we are working with the expression \( \log _{5}\left(\frac{1}{25}\right) \), so we need to find the power that 5 must be raised to get \( \frac{1}{25} \).

Step 2: Express 'x' as a Power of the Base

We know that \( 5^2 = 25 \). Therefore, \( \frac{1}{25} = \frac{1}{5^2} \). This can be rewritten as \( 5^{-2} \).

Step 3: Evaluate the Logarithmic Expression Using the Exponent

Now we substitute \( \frac{1}{25} \) with \( 5^{-2} \) in the logarithmic expression: \( \log _{5}(5^{-2}) \). According to logarithmic rules, \( \log _{b}(b^{x}) = x \). Thus, \( \log _{5}(5^{-2}) = -2 \).

Key Concepts

Exponents

Exponents are a fundamental mathematical concept used to represent repeated multiplication. When you see a number written as a base with an exponent, like \(b^n\), it simply means the base \(b\) is multiplied by itself \(n\) times.
Exponents can also be used inversely, indicating division when negative, or fractional exponents symbolizing roots.

• Positive exponents: Indicate how many times a number is multiplied by itself, such as \(2^3 = 2 \times 2 \times 2 = 8\).
• Negative exponents: Represent the reciprocal of the base raised to the opposite positive power, like \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\).
• Zero exponent: Any non-zero number raised to the power of zero equals 1, such as \(7^0 = 1\).

Understanding exponents helps greatly in simplifying logarithmic expressions, such as transforming \(\frac{1}{25}\) into \(5^{-2}\). This step is crucial for solving problems that involve logarithms.

Logarithms

Logarithms are the inverse operation of exponentiation, letting us solve for the power a base must be raised to achieve a certain number. The notation \(\log_b(x)\) asks, "What power should the base \(b\) be raised to get \(x\)?"
In problems like \(\log_{5}(\frac{1}{25})\), understanding this concept is key to finding solutions.

• Definition: If \(b^y = x\), then \(\log_b(x) = y\).
• Base and Argument: 'Base' \(b\) in \(\log_b(x)\), is the number to be raised, and 'Argument' \(x\) is the result from raising the base to a power.
• Logarithmic Identity: \(\log_b(b^y) = y\), meaning the log of a base raised to a power will equal that power.

By converting numbers into base powers, particularly with exponents, you can compute logarithmic expressions effectively, like simplifying \(\log_{5}(5^{-2})\) to directly get \(-2\).

Base Conversion

Base conversion in context of logarithms and exponentials involves changing the form or perception of numbers to simplify computation. This especially applies when finding equivalent forms using different bases can make calculations easier.
Through conversion, we can also verify results or solve problems involving non-standard bases.

• Identifying Power Forms: Convert fractions or composite numbers to exponential forms relative to desired bases, as \(\frac{1}{25}\) to \(5^{-2}\), aligning with base-5.
• Using Base Conversion: Changing logarithms to different bases can solve more complex equations or could be specified by problems requiring a common base. For example, using the change of base formula: \(\log_b(x) = \frac{\log_c(x)}{\log_c(b)}\).
• Verification: Base conversion can help check the accuracy of solutions, offering a method to reassess tasks and confirm correctness.

This conversion principle aids students not only in single expressions but also in acknowledging diverse paths to solving exponential or logarithmic challenges.