Problem 20 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{1} 5^{4}} & {\text { (b) } \log _{4} 64} & {\text { (c) } \log _{1} 9}\end{array} $$

Step-by-Step Solution

Answer

(a) Undefined, (b) 3, (c) Undefined.

Step 1: Evaluate \( \log_1 5^4 \)

The expression \( \log_a b \) is only defined when \( a > 0 \) and \( a eq 1 \). Since the base here is 1, the expression \( \log_1 5^4 \) is undefined.

Step 2: Simplify \( 64 \text{ as a power of } 4 \)

To evaluate \( \log_4 64 \), we need to express 64 as a power of 4. Recall that \( 4 = 2^2 \), therefore, \( 64 = (2^2)^3 = 2^6 = (4^3) \). So, \( 64 \) can be written as \( 4^3 \).

Step 3: Evaluate \( \log_4 64 \)

We know that if \( b = a^c \), then \( \log_a b = c \). Since \( 64 = 4^3 \), \( \log_4 64 = 3 \).

Step 4: Evaluate \( \log_1 9 \)

Similar to part (a), \( \log_a b \) requires \( a > 0 \) and \( a eq 1 \). Since the base is 1, \( \log_1 9 \) is also undefined.

Key Concepts

Understanding the Base of Logarithm

The base of a logarithm is a fundamental part of understanding how logarithmic operations work. Logarithms are denoted by \( \log_a b \), where \( a \) is the base. This signifies how many times the base \( a \), must be multiplied by itself to get \( b \).
Logarithms follow specific guidelines for their base:

• The base \( a \) must be greater than 0.
• The base \( a \) must not be equal to 1.

These rules ensure that the logarithmic expressions are well-defined and meaningful. If the base is 1, as in the examples \( \log_1 5^4 \) and \( \log_1 9 \), the logarithm expression becomes undefined. The concept of the base being greater than zero and not equal to one prevents undefined expressions and ensures the mathematical operation is valid.
Logarithm with suitable base is a powerful tool in mathematics and it is crucial to always check these conditions when evaluating logarithmic expressions.

Exploring Undefined Logarithms

An undefined logarithm occurs when the conditions for the logarithmic base are not met. Specifically, if the base \( a \) is less than or equal to 0, or exactly equal to 1, the logarithm cannot be determined.
Let’s consider the expression \( \log_1 5^4 \). Here, the potential base is 1, which does not adhere to the required criteria of \( a > 0 \) and \( a eq 1 \). This is why \( \log_1 5^4 \) is classified as undefined, meaning it lacks a valid numerical answer.
Similarly, in \( \log_1 9 \), the base is again 1. This makes the logarithm undefined for the same reason as above. It is essential to verify not only the values but specifically how bases are set when dealing with logarithmic expressions to avoid undefined scenarios.
By recognizing why certain logarithmic forms are undefined, students can better anticipate errors and ensure their calculus follows the mathematical rules.

Logarithmic Expression Evaluation Made Easy

Evaluating a logarithmic expression involves breaking it down into manageable steps. Let’s take the example \( \log_4 64 \).
First step is expressing the number within the logarithm, in this case 64, as a power of the base 4. Knowing that 64 equals \( 4^3 \), because \( 64 = (4)^3 \), sets clear grounds for evaluation.
After establishing this, the actual logarithm evaluation becomes straightforward:

• If \( b = a^c \), then \( \log_a b = c \). In our case, since \( 64 = 4^3 \), this means \( \log_4 64 = 3 \).

By reducing the number to its base's power, we can see clearly the answer appears from such relationship between exponents and logarithms. This not only simplifies calculations but also provides a deeper understanding of how logarithms function and relate to exponential equations.
For even greater clarity, always break these steps down, ensuring each part of the expression relates back to its fundamentals in math basics.