Problem 91 - Algebra 2

Question

Which statement is NOT correct? \(\begin{array}{ll}{\text { A. } \log _{2} 25=2 \cdot \log _{2} 5} & {\text { B. } \log _{3} 16=2 \cdot \log _{3} 8} \\ {\text { C. } \log _{5} 27=3 \cdot \log _{5} 3} & {\text { D. } \log _{8} 10,000=4 \cdot \log _{8} 10}\end{array}\)

Step-by-Step Solution

Answer

Statement B is the one that is not correct.

Step 1: Checking statement A

For statement A, it is expressed as \(\log_2 25 = 2 \cdot \log_2 5\). This is true when checking against our logarithmic rules since \( 2^2 = 4 \) and \( 5^2 = 25 \) making statement A correct.

Step 2: Checking statement B

For statement B, \(\log_3 16 = 2 \cdot \log_3 8\). This is not true because \(3^2 = 9\) and \(8^2 = 64\). Thus, statement B is false which means it's the answer as it's the statement that is not correct.

Step 3: Checking statement C

For the purpose of completeness, we will also test statement C: \(\log_5 27 = 3 \cdot \log_5 3\). This is true because \(5^3 = 125\) and \(3^3 = 27\), which makes C correct.

Step 4: Checking statement D

Testing statement D: \(\log_8 10,000 = 4 \cdot \log_8 10\). Just like the checks above, this matches with the rule because \(8^4 = 4096\) and \(10^4 = 10,000\), making statement D as well correct.

Key Concepts

Logarithmic Rules

Logarithmic rules are essential for simplifying complex expressions with logarithms. One of the core rules is the power rule, which states

• \( \log_b(m^n) = n \cdot \log_b(m) \)

This allows you to move exponents in the argument of a logarithm to be coefficients outside the logarithm. For example, in statement A from the exercise \( \log_2 25 = 2 \cdot \log_2 5 \), using the power rule, 25 can be expressed as \( 5^2 \). Therefore, the equation becomes \( \log_2 (5^2) = 2 \cdot \log_2 5 \), which verifies the correctness of the statement.

Understanding and applying these rules allow you to simplify and manipulate logarithmic expressions efficiently.

Properties of Exponents

Exponents often work hand-in-hand with logarithms, as seen in the logarithmic rules. Exponentiation has several fundamental properties:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
• Power of a Product: \( (a\cdot b)^n = a^n \cdot b^n \)

These properties are important in the context of logarithms. For instance, checking statement C, \( \log_5 27 = 3 \cdot \log_5 3 \), requires recognizing that 27 is a product of threes (i.e., \( 3^3 \)), aligning with the power of a product property. This understanding reinforces why certain logarithmic transformations hold true in mathematical reasoning.

Mathematical Reasoning

Mathematical reasoning is crucial in problem-solving, enabling you to discern the validity of statements. In the exercise, applying reasoning involves checking each statement against known mathematical principles, such as logarithmic rules and exponent properties.

For statement B, where \( \log_3 16 = 2 \cdot \log_3 8 \) is evaluated, reasoning shows it is incorrect when compared against the rules of logarithms and exponents. By understanding that \(3^2\) is 9 and \(8^2\) is 64, you immediately see that the statement doesn't hold, showcasing mathematical reasoning's role in validating or refuting expressions.

Therefore, using logical reasoning and step-by-step verification ensures accurate results and clarifies complex mathematical concepts.