Problem 18 - Intermediate Algebra

Question

Complete the table of values. $$ f(x)=\log _{8} x $$ $$ \begin{array}{|c|c|} \hline \text { Input } & \text { Output } \\ \hline 8 & \\ \hline-8 & \\ \hline 0 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

When \( x = 8 \), Output is 1; for \( x = -8 \) and \( x = 0 \), Output is undefined.

Step 1: Understand the Logarithm

The function given is \( f(x) = \log_{8}x \), which represents the logarithm of \( x \) with base 8. This function answers the question: `To what power must 8 be raised, to produce \( x \)`? The general form is \( y = \log_{b}x \) and translates to \( b^y = x \).

Step 2: Calculate Output for \( x = 8 \)

For \( x = 8 \), we are finding \( \log_{8}8 \). Since 8 to the power of 1 is 8 (i.e., \( 8^1 = 8 \)), we have \( \log_{8}8 = 1 \). Thus, the Output is 1.

Step 3: Consider Input \( x = -8 \)

The logarithm of a negative number is undefined in the real number system because a positive base raised to any real number power cannot yield a negative number. Thus, \( \log_{8}(-8) \) is undefined.

Step 4: Calculate Output for \( x = 0 \)

The logarithm of zero is undefined in the real number system because no power of a positive number like 8 can be zero. Thus, \( \log_{8}(0) \) is also undefined.

Key Concepts

Base of a Logarithm

The base of a logarithm is an important concept. It tells us which number is being raised to a power to get another number. In the function \( f(x) = \log_{8}x \), the number 8 is the base. This means we're asking: "To what power do we raise 8 to get \( x \)?" Let's break it down further:

• Base Number: The value 8 in \( \log_{8}x \) is known as the base. It's the number that is repeatedly multiplied in the operation.
  
• Power: In the expression, \( \log_{8}x = y \), the power is \( y \). It represents how many times the base is multiplied by itself to reach \( x \).
  
• Equation: The equation \( b^y = x \) demonstrates this relationship. Here, base \( b \) (which is 8) raised to some power \( y \) equals \( x \).

Choosing a base is crucial because it determines the growth pattern and function behavior. Different bases lead to different logarithmic scales.

Logarithm of Negative Numbers

Logarithms don't mix well with negative numbers when we stick to the real numbers. This is because positive numbers, such as our base 8, cannot be raised to any real power that results in a negative number.

• Concept of Negative Logarithm: In math, the logarithm asks how many times we multiply the base by itself to reach another number. But since multiplying a positive number by itself always results in another positive number, we simply can't get a negative.
  
• Example: Imagine trying to answer \( \log_{8}(-8) \). Our function asks, "What power do we raise 8 to make it -8?" Since there's no real power that lets 8 result in -8, the answer is undefined.
  
• Conclusion: With logarithms, keep the argument (that’s the number you're taking the log of) positive. It's only when we step into complex numbers, a more advanced field, that logarithms of negatives become defined. For now, in real numbers, we say it's undefined.

Undefined Logarithms

Sometimes logarithms can be undefined. This happens if their calculations break certain mathematical rules. A common situation in which this occurs is when trying to find the logarithm of zero.

• Logarithm of Zero: Consider \( \log_{8}(0) \). We’re asking what power of 8 will get us 0. Since any positive number raised to any power remains positive, it’s impossible to equal 0. Therefore, \( \log_{8}(0) \) is undefined in the realm of real numbers.
  
• Understanding Undefined: Whenever you see a logarithm with zero or a negative number, check if you’re working in real numbers. Here, such scenarios are always undefined because they can’t satisfy the exponential equations involved.
  
• Importance: Recognizing undefined logarithms helps avoid miscalculations. It's essential, especially when solving problems, to ensure only valid operations are used.