Problem 71 - Algebra 2

Question

Find the expression that does not belong. Explain. \(\log _{4} 16\) \(\qquad\) \(\log _{2} 16\) \(\qquad\) \(\log _{2} 4\) \(\qquad\) \(\log _{3} 9\)

Step-by-Step Solution

Answer

The expression \( \log_2 16 \) does not belong because its value is 4, unlike the others, which have the value 2.

Step 1: Calculate Each Logarithm

First, let's calculate each logarithm to find their values. - \( \log_4 16 = 2 \)- \( \log_2 16 = 4 \)- \( \log_2 4 = 2 \)- \( \log_3 9 = 2 \)This step involves finding how many times the base must be multiplied by itself to result in the number.

Step 2: Compare the Values

Next, compare the values obtained from the previous step:- \( \log_4 16 = 2 \)- \( \log_2 16 = 4 \)- \( \log_2 4 = 2 \)- \( \log_3 9 = 2 \)Notice that the values \( \log_4 16 \), \( \log_2 4 \), and \( \log_3 9 \) are equal to 2, while \( \log_2 16 \) is equal to 4.

Step 3: Identify the Outlier

Identify which expression does not have the same value as the others. In this case, the outlier is:- \( \log_2 16 = 4 \)This is because it is distinct from the other values, which are all equal to 2.

Key Concepts

Logarithmic expressions

Logarithms provide an inverse operation to exponentiation, helping us solve equations where the unknown is in the exponent. At its core, a logarithmic expression involves a base and a number, answering the question, "To what exponent must the base be raised to produce this number?" When you see an expression like \(\log_b a\), it reads as "log base \(b\) of \(a\)" and finds the power to which \(b\) must be raised to yield \(a\).

Understanding this is essential for evaluating logarithmic expressions, as it forms the basis of interpreting the results. When we calculated logarithms in the exercise, like \( \log_4 16 \), we were determining how many times the base (4) needs to be multiplied by itself to get 16. The answer was 2, meaning \(4^2 = 16\).

In simpler terms:

• \(\log_4 16 = 2\) because \(4^2 = 16\).
• \(\log_2 16 = 4\) because \(2^4 = 16\).
• \(\log_2 4 = 2\) because \(2^2 = 4\).
• \(\log_3 9 = 2\) because \(3^2 = 9\).

Grasping these basics allows you to analyze and compare logarithmic expressions effectively.

Evaluating logarithms

"Evaluating logarithms" simply involves finding the value of the logarithm, which answers the fundamental question raised by logarithmic expressions: To what power must the base be raised to produce a specific number? The step-by-step solution in the exercise demonstrates this evaluation process.

Let's break down this methodology:

• Identify the base and the number in your expression.
• Determine how many times you must multiply the base by itself to get the number.
• Write down the result, which is the power needed for the base to reach the number.

For example, when evaluating \(\log_4 16\), the base is 4, and the number is 16. You seek the power of 4 that returns 16, which is 2, hence \(\log_4 16 = 2\). Each time you evaluate a logarithm, you pinpoint how many exponent steps are needed to transform the base into the number.

Evaluating logarithms is not just about memorization but understanding these relative powers and how they interact.

Properties of logarithms

Logarithms follow specific properties that simplify calculations, making it easier to handle complex expressions without always resorting to step-by-step evaluations. These properties are rooted in the rules of exponents and can help identify unique relationships between logarithmic expressions.

Some essential properties are:

• Product Property: \(\log_b (mn) = \log_b m + \log_b n\)
• Quotient Property: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\)
• Power Property: \(\log_b (m^n) = n \cdot \log_b m\)
• Change of Base Formula: \(\log_b a = \frac{\log_k a}{\log_k b}\), useful for converting bases.

In the exercise, recognizing that \(\log_4 16\), \(\log_2 4\), and \(\log_3 9\) all equate to 2 aligns with their base-2 relationships. The property that sticks out is that \(\log_2 16 = 4\), which does not share the same resonant power, making it an outlier. Understanding these properties equips you to discern inconsistencies or patterns within sets of logarithms without directly solving each one.