Problem 50 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{8} 8^{k}$$

Step-by-Step Solution

Answer

\(k\)

Step 1: Recognize the Logarithmic Identity

Start by recognizing that there is a logarithmic identity we can use. The identity states that if you have the logarithmic expression \(\log_{b}(b^{x})\), it simplifies to \(x\). This is because you are essentially asking, 'to what power must \(b\) be raised to get \(b^{x}\)?' The answer is simply \(x\).

Step 2: Apply the Identity to the Given Expression

Now, apply the identity from Step 1 to the expression we need to simplify: \(\log_{8}(8^k)\). According to the identity, since the base of the log function (8) matches the base of the exponent (8), the expression simplifies directly to \(k\).

Step 3: Conclusion: Simplified Result

Based on the above steps, the expression \(\log_{8}(8^{k})\) simplifies directly to \(k\). There are no additional calculations required.

Key Concepts

Logarithmic Identity

When simplifying expressions involving logarithms, recognizing and using logarithmic identities is highly beneficial. One powerful logarithmic identity is the one used when dealing with an expression of the form \( \log_{b}(b^{x}) \). This identity states that \( \log_{b}(b^{x}) = x \). It might seem too good to be true at first glance, but it perfectly illustrates the nature of logarithms as inverse operations of exponentiation.

To break it down: the logarithm is essentially asking, "To what power must the base \( b \) be raised, to produce \( b^{x} \)?" The answer is straightforward—it is simply \( x \). When you see such an expression, this identity allows for immediate simplification, eliminating the need for extensive calculations.

• If the bases of the logarithm and the exponent match, the exponent becomes the answer directly.
• This identity helps to streamline mathematical processes by cutting down unnecessary steps.

Logarithmic Expression

A logarithmic expression involves a logarithm, which is a math function that helps to answer questions related to the power a number must be raised to achieve another number. It typically takes the form of \( \log_{b}(x) \), where \( b \) is the base of the logarithm, and \( x \) is the argument. Analyzing and simplifying logarithmic expressions is a common task in algebra and beyond.

Understanding the components:

• **Base**: This is the number you raise to a power.
• **Argument**: The result you achieve by raising the base to some power.

Simplifying a logarithmic expression means finding a way to rewrite it in a simpler or more straightforward form using logarithmic identities or rules, such as the product rule, quotient rule, or power rule. In the case of \( \log_{8}(8^{k}) \), the expression is simplified using the aforementioned logarithmic identity as it's a perfect match for it.

Knowing these simplification techniques is crucial when dealing with exponential and logarithmic problems in various mathematical fields.

Base of a Logarithm

The base of a logarithm is a fundamental concept that gives structure and meaning to a logarithmic function. The base \( b \) in \( \log_{b}(x) \) indicates the number we repeatedly multiply to achieve the argument \( x \). The choice of base can dramatically change the nature and result of the logarithm.

• A common base is \(10\), often called the "common logarithm."
• The base \(e\) (approximately 2.718) is termed the "natural logarithm."
• Logarithms with base 2 are frequently used in computer science.

In our exercise, \(8\) is the base. Understanding the role of the base in a logarithmic expression is key to mastering logarithms. In cases where the base of the logarithm matches the base of the exponent, ideally, this leads to a quick shortcut via the identity \( \log_{b}(b^{x}) = x \), significantly simplifying complex expressions. Understanding these nuances will aid in unraveling more complicated mathematical problems involving logarithms.