Problem 40 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{10} \frac{1}{10} $$

Step-by-Step Solution

Answer

The value is -1.

Step 1: Understand the Logarithmic Expression

The expression given is \( \log_{10} \frac{1}{10} \). This notation represents the logarithm with base 10 of the fraction \( \frac{1}{10} \).

Step 2: Identify the Base and Argument

In \( \log_{10} \frac{1}{10} \), the base of the logarithm is 10, and the argument of the logarithm is \( \frac{1}{10} \).

Step 3: Recall the Logarithmic Rule

Remember the rule \( \log_{b}(\frac{1}{b}) = -1 \). This rule states that the logarithm of a reciprocal of the base is \(-1\).

Step 4: Apply the Logarithmic Rule

Using the rule \( \log_{10} \frac{1}{10} = -1 \), apply it directly since the argument is the reciprocal of the base.

Key Concepts

Logarithmic Rules

Logarithms have a set of rules that simplify the process of manipulating and understanding logarithmic expressions. One key rule is that \(\log_{b}(b^x) = x\). This means that whenever you take the logarithm of a power of the same base, it simplifies to just the power itself.
For example, \(\log_{10}(10^2) = 2\).
Another important rule is the product rule: \(\log_{b}(MN) = \log_{b}(M) + \log_{b}(N)\).
This tells us that the logarithm of a product is the sum of the logarithms of the multiplicands.

• Quotient Rule: \(\log_{b}\left(\frac{M}{N}\right) = \log_{b}(M) - \log_{b}(N)\)
• Reciprocal Rule: \(\log_{b}\left(\frac{1}{M}\right) = -\log_{b}(M)\)
• Change of Base Formula: \(\log_{b}(M) = \frac{\log_{k}(M)}{\log_{k}(b)}\)

Understanding these rules is crucial when solving logarithmic equations, as they allow for the simplification and transformation of complex expressions into more manageable forms. The reciprocal rule is particularly useful, as seen in the exercise, where \(\log_{10}\frac{1}{10} = -1\).

Base of Logarithm

The base in a logarithm is a foundational element that defines the scale of measurement used in the logarithmic expression. In the expression \(\log_{b}x\), "\(b\)" is the base.
A common base is 10, often referred to as the "common logarithm," denoted simply as \(\log\). This base aligns with our decimal system, making it practical for many calculations.
The base "e," approximately equal to 2.718, is used in natural logarithms, denoted as \(\ln\). It appears frequently in continuous growth processes, such as population dynamics.

• The base helps determine what power the base must be raised to in order to achieve the given number, or argument.
• While bases like 10 and e are frequent, any positive number (except 1) can serve as a base in logarithms.

Understanding the base is crucial in making sense of logarithms, as it provides context to the comparison between the base and the argument (the number within the logarithm). In our exercise, knowing the base was critical for simplifying the expression \(\log_{10}\frac{1}{10}\).

Reciprocal in Logarithms

Understanding the reciprocal in logarithmic contexts can greatly simplify solving logarithmic expressions. The reciprocal of a number \(x\) is \(\frac{1}{x}\). When this concept is applied to logarithms, it connects beautifully with the logarithmic rules.
The specific rule that relates to reciprocals is \(\log_{b}\left(\frac{1}{M}\right) = -\log_{b}(M)\).
This indicates that the logarithm of a reciprocal of a positive number is the negative of the logarithm of the number itself.

• This property is derived from the fact that a reciprocal indicates an inverse, and inverses in logarithms correlate to negative values.
• This simplifies the calculation significantly, saving steps and reducing the potential for errors.

For instance, in our exercise, by identifying \(\frac{1}{10}\) as the reciprocal of 10, we directly applied the reciprocal rule to arrive at \(\log_{10}\frac{1}{10} = -1\). Recognizing and utilizing this rule can provide an efficient pathway to solve similar logarithmic problems.