Problem 38 - Contemporary Precalculus

Question

Compute each of the following pairs of numbers. (a) \(\log 18\) and \(\frac{\ln 18}{\ln 10}\) (b) \(\log 456\) and \(\frac{\ln 456}{\ln 10}\) (c) \(\log 8950\) and \(\frac{\ln 8950}{\ln 10}\) (d) What do these results suggest?

Step-by-Step Solution

Answer

Answer: The relationship between the logarithm base 10 and the natural logarithm values for a given number is that \(\log x = \frac{\ln x}{\ln 10}\). This relationship, known as the change of base formula, allows us to convert between the two types of logarithms.

Step 1: (a) Compute \(\log 18\) and \(\frac{\ln 18}{\ln 10}\)

First, calculate the logarithm base 10 of 18: \(\log 18 \approx 1.255\) Next, calculate the natural logarithm of 18 and divide it by the natural logarithm of 10: \(\frac{\ln 18}{\ln 10} \approx \frac{2.890}{2.303} \approx 1.255\)

Step 2: (b) Compute \(\log 456\) and \(\frac{\ln 456}{\ln 10}\)

First, calculate the logarithm base 10 of 456: \(\log 456 \approx 2.659\) Next, calculate the natural logarithm of 456 and divide it by the natural logarithm of 10: \(\frac{\ln 456}{\ln 10} \approx \frac{6.122}{2.303} \approx 2.659\)

Step 3: (c) Compute \(\log 8950\) and \(\frac{\ln 8950}{\ln 10}\)

First, calculate the logarithm base 10 of 8950: \(\log 8950 \approx 3.952\) Next, calculate the natural logarithm of 8950 and divide it by the natural logarithm of 10: \(\frac{\ln 8950}{\ln 10} \approx \frac{9.100}{2.303} \approx 3.952\)

Step 4: (d) Observing the results

Observing the results in each part (a, b, and c), we can see that the values for \(\log x\) and \(\frac{\ln x}{\ln 10}\) are equal or approximately equal for each given number 'x'. This suggests that: \(\log x = \frac{\ln x}{\ln 10}\) This relationship holds true because the logarithm base 10 of a number can be converted into a natural logarithm by dividing the natural logarithm of the number by the natural logarithm of 10. This observation is a property of logarithms called the change of base formula.

Key Concepts

Change of Base Formula

The change of base formula is a vital tool in mathematics that allows us to convert logarithms from one base to another. This formula is particularly useful when you need to compute logarithms in a different base than what a calculator or software provides. It states: \[ \log_b x = \frac{\ln x}{\ln b} \] In the context of our exercise, we often deal with converting from other bases to base 10. For example, to convert from the natural logarithm (base \(e\)), we use: \[ \log_{10} x = \frac{\ln x}{\ln 10} \] This equivalence demonstrates that logarithms calculated using natural logs can be transformed to base 10 using this straightforward method. The change of base formula simplifies calculations, ensures consistency across different contexts and tools, and offers a smooth transition between logarithmic systems.

Natural Logarithm (ln)

The natural logarithm, denoted as \(\ln\), is a logarithm with base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. The natural logarithm is widely used due to its mathematical properties and its appearance in various natural growth processes and phenomena. The expression \(\ln x\) represents the power to which the base \(e\) must be raised to yield the number \(x\). For instance, \(\ln e = 1\) because \(e^1 = e\).

• It is commonly used in calculus, particularly for solving differential equations and in integral calculus.
• In finance, economics, and biology, \(\ln\) models continuous growth processes, such as compound interest or population dynamics.

Understanding \(\ln\) aids in grasping more complex mathematical concepts and enhances the ability to model real-world phenomena accurately.

Logarithm Base 10 (log)

Logarithms with base 10 are called common logarithms and are typically denoted as \(\log\). These are frequently used in scientific fields because they correspond with the decimal numbering system, simplifying calculations involving powers of ten. The expression \(\log x\) indicates the power to which 10 must be raised to produce \(x\). For example, \(\log 100 = 2\) because \(10^2 = 100\).

• They are often employed in fields like chemistry, physics, and engineering, where systems are analyzed in terms of orders of magnitude.
• They provide a way to express very large or very small numbers compactly, such as pH levels in chemistry or sound levels in decibels.

The common logarithm's association with base 10 makes it intuitive when working within the base-10 numeric system.

Mathematical Computation

Mathematical computation involves various techniques and methods used to solve problems, from simple arithmetic to complex calculus. It plays a crucial role in both pure mathematics and applied sciences. The computations involved in solving the given logarithmic expressions show practical applications of these mathematical concepts.

• Using a calculator or computational software simplifies the process of evaluating logarithmic expressions.
• Understanding the underlying principles, such as the change of base formula, ensures that students can confidently transition between different types of logarithms and apply them across different scenarios.

Accurately performing mathematical computations enhances problem-solving abilities and ensures precise results, which are essential for academic success and various professional fields. By practicing these calculations, learners can solidify their understanding and apply these skills effectively in both exams and real-life situations.