Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

Is \(\log _{3} 24=\frac{\ln 3}{\ln 24}\) or \(\log _{3} 24=\frac{\ln 24}{\ln 3}\) correct?

Step-by-Step Solution

Answer

\(\log _{3} 24 = \frac{\ln 24}{\ln 3}\) is the correct representation.

Step 1: Express \(\log _{3} 24\) in Natural Logarithms

Use the change of base formula to rewrite \(\log _{3} 24\) in terms of natural logarithms. Thus, \(\log _{3} 24 = \frac{\ln 24}{\ln 3}\)

Step 2: Compare the Result with the Given Options

Compare the resulting expression \(\frac{\ln 24}{\ln 3}\) with the given options. It can be clearly seen that \(\log _{3} 24 = \frac{\ln 24}{\ln 3}\) and \(\log _{3} 24 = \frac{\ln 3}{\ln 24}\) corresponds to the original statement and the result obtained, respectively.

Step 3: Draw the Conclusion

Based on the comparison, it can be concluded that \(\log _{3} 24 = \frac{\ln 24}{\ln 3}\) is correct, while \(\log _{3} 24 = \frac{\ln 3}{\ln 24}\) is incorrect.

Key Concepts

Natural Logarithms

Natural logarithms are a type of logarithmic function that use the number 'e' (approximately 2.71828) as their base. The natural logarithm of a positive number 'x' is generally written as \(\ln x\) and represents the power to which 'e' must be raised to obtain the number 'x'.

Natural logarithms are particularly important in mathematics due to their appearance in many natural phenomena, including compound interest, population growth, and radioactive decay. They form the basis of calculus, appearing in the integration of many functions and in solving differential equations.

Understanding natural logarithms is crucial for grasping the relationship between different logarithmic scales and for converting between them, which brings us to the significance of the change of base formula—used to re-express logarithms from one base to another using natural logarithms.

Logarithmic Expressions

A logarithmic expression is a way to represent exponential relationships. The notation \(\log_b x\) denotes the logarithm of 'x' with base 'b'. The foundation of logs lies in the fact that logarithms and exponents are inverses of each other; to put it simply, \(\log_b (b^y) = y\) and \(b^{(\log_b x)} = x\).

When working with logarithmic expressions, it's vital to remember key properties, such as the product rule \(\log_b (xy) = \log_b x + \log_b y\), the quotient rule \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\), and the power rule \(\log_b (x^y) = y\log_b x\). These rules simplify the process of solving logarithmic equations or changing the base of logarithmic expressions.

Moreover, the change of base formula is a pivotal tool used in shifting from one logarithmic base to another, making it possible to solve problems when a different base is more convenient or when using a calculator that only has buttons for \(\log\) (base 10) or \(\ln\) (natural logarithm).

Mathematical Problem-Solving

Mathematical problem-solving encompasses understanding the problem, developing a plan, executing that plan, and then revisiting the solution for accuracy—a strategy known as George Pólya's four-step problem-solving process. This structured approach can be applied to complex tasks like evaluating logarithmic expressions.

Key to successful problem-solving in mathematics is a solid grasp of the underlying concepts. For instance, with logarithms, remembering that \(\log_b \left(\frac{A}{B}\right) = \log_b A - \log_b B\) aids in recognizing why \(\log _{3} 24 = \frac{\ln 24}{\ln 3}\) rather than \(\frac{\ln 3}{\ln 24}\). Careful comparison of expressions, as in the step by step solutions, also showcases the systematic approach needed to correctly solve and understand mathematical expressions.

Moreover, cultivating skills in manipulating logarithmic and exponential expressions, understanding the change of base formula, and the impact of different bases are essential tools in a student's mathematical toolkit for efficient problem-solving.