Problem 76 - Algebra A Combined Function

Question

Simplify. $$ \log _{11}(11)^{-1} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-1\).

Step 1: Understanding the Problem

The expression to simplify is \( \log_{11}(11)^{-1} \). Here, we have a logarithm with base 11 and the argument \((11)^{-1}\). The goal is to simplify this expression.

Step 2: Using Logarithm Property

Apply the logarithm power rule: \( \log_b (a^c) = c \cdot \log_b a \). In our case, \(a = 11\), \(c = -1\), and \(b = 11\). Therefore, \( \log_{11}(11)^{-1} = -1 \cdot \log_{11} 11\).

Step 3: Simplifying Further

Recognize that \( \log_b b = 1 \) for any positive \(b\). Hence, \( \log_{11} 11 = 1 \). Substitute back to get: \(-1 \cdot 1 = -1\).

Step 4: Conclusion

Thus, the simplified form of the expression \( \log_{11}(11)^{-1} \) is \(-1\).

Key Concepts

Logarithm Properties

Logarithms have several useful properties that help in simplifying expressions, such as the power rule, product rule, and quotient rule.
In our exercise, we use the power rule, which states:

• \( \log_b (a^c) = c \cdot \log_b a \)

This property allows us to take the exponent out of the logarithm as a multiplying factor. It's particularly helpful when you're dealing with powers.

By applying this to the expression \( \log_{11}(11)^{-1} \), we recognize \( a = 11 \), \( c = -1 \), and \( b = 11 \). Therefore, using the power rule, the expression becomes \(-1 \cdot \log_{11} 11\).
This brings us to another basic property:

• The identity \( \log_b b = 1 \).

Understanding these properties is key to simplifying any logarithmic expression efficiently.

Base of a Logarithm

The base of a logarithm is a critical component because it tells us the number in terms of which we are expressing the power. For example, in \( \log_{11}(11)^{-1} \), 11 is the base.

The base is always a positive number, except the special case where the logarithm base \(e\) leads to natural logarithms. In many logarithmic expressions, recognizing the base can make simplification much easier.

• For instance, if the base and the argument (the number next to the base) are the same, then by property \( \log_b b^c = c \), it simplifies the process greatly.
• Another example is \( \log_b b \), which equals to 1. This identity is exactly what's used when simplifying \( \log_{11} 11 \).

By understanding which base you have, and the properties that relate to it, you can simplify many logarithmic expressions quickly and correctly.

Simplifying Expressions

Simplifying logarithmic expressions is an important skill that allows you to work with and transform logarithmic identities and equations into simpler forms.

To simplify \( \log_{11}(11)^{-1} \): start with identifying properties of logarithms, such as the power rule and base-argument relationships. Use these properties:

• Apply the power rule: \( \log_b (a^c) = c \cdot \log_b a \). In this case, our expression becomes \(-1 \cdot \log_{11} 11\).
• Use the identity \( \log_{11} 11 = 1 \), derived from \( \log_b b = 1 \). Now your expression simplifies further to \(-1 \cdot 1\), which results in \(-1\).

By understanding each step and applying these rules methodically, we achieve a neat, simplified form of the given logarithmic expression. Such simplifications are crucial in higher mathematics to solve equations or evaluate expressions effectively.