Problem 123 - College Algebra Essentials

Question

Determine whether statement makes sense or does not make sense, and explain your reasoning. I can use any positive number other than 1 in the change-of-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

Step-by-Step Solution

Answer

The statement makes sense because while the change-of-base formula allows for any positive number to be used, most calculators only provide logarithmic calculations for base 10 and \(e\), making the use of these bases more practical.

Step 1: Interpret the Statement

The given statement says that any positive number can be used as a base in the change-of-base property for logarithms, but it's most practical to use base 10 or base \(e\) because calculators typically offer these bases.

Step 2: Understanding of Change-of-Base Property

The statement refers to the change-of-base property of logarithms, which states that the base of a logarithm can be changed to any other base using the formula: \(\log_b a = \frac{\log_d a}{\log_d b}\), where \(d\) can technically be any positive number besides 1 according to the property.

Step 3: Application of statement to practical situations

While it's true that the formula allows for \(d\) to be any positive number other than 1, the given statement argues that bases 10 and \(e\) are most practical, which has to do with how most scientific calculators are designed. Most calculators, including scientific ones, often provide logarithms for bases 10 (common logarithms) and \(e\) (natural logarithms). Therefore in a practical scenario, using these base values tends to be more convenient.

Key Concepts

Logarithms

Logarithms are the inverse operation to exponentiation, meaning they help us find the power to which a base number must be raised to get another number. For example, if you have \(10^2 = 100\), the logarithm tells you that

• \(\log_{10}{100} = 2\)

This operation is particularly useful in solving exponential equations or dealing with exponential growth, such as in population studies or financial compounding.
Another point to note is that the logarithm's base must be a positive number except one, as a base of one makes the logarithmic operation undefined.
The true power of logarithms comes from properties like the change-of-base formula, allowing you to switch bases, such as to base 10 or base \(e\), making calculations easier with the tools available, like calculators.

Base 10

The base 10 logarithm, also known as the common logarithm, is frequently used due to its straightforward nature. With base 10, you often hear it associated with terms like decibels, pH levels, and the Richter scale. The base 10 logarithm is written as \(\log_{10}{(x)}\) or simply \(\log{(x)}\).
Using the change-of-base property, you can express any logarithm in terms of base 10:

• \(\log_b{a} = \frac{\log_{10}{a}}{\log_{10}{b}}\)

This property is helpful when using calculators that default to base 10 functions, making it easier to solve problems without complex calculations.

Base e

Base \(e\), represented mathematically as \(\log_{e} (x)\) and often notated as \(\ln(x)\), stands for natural logarithms. The number \(e\) itself is an irrational and transcendental number approximately equal to 2.718. Natural logarithms are widely used in higher mathematics and science, especially in fields like calculus and probability.
The change-of-base formula provides a way to convert base \(e\) logarithms into any other base you might be working with, but most scientific calculators allow fast computation of natural logs, making them user-friendly for scientific and engineering applications.

• They are critical in modeling real-world processes like growth and decay, where quantities change exponentially over time.
• Using \(e\) simplifies many mathematical expressions found in calculus and analysis.

Scientific Calculators

Scientific calculators are versatile tools designed to handle a variety of mathematical operations, including those involving logarithms. These calculators are typically programmed to quickly and easily compute both base 10 (common) and base \(e\) (natural) logarithms without needing further transformations.

• They're essential tools in many fields, making logarithmic calculations straightforward and efficient for students, scientists, and engineers alike.
• They eliminate the need for memorizing complex formulas by offering direct functions, simplifying tasks such as solving exponential equations.

In practical applications, using these calculators saves time and reduces errors in calculations related to logarithms. This is why having an understanding of base 10 and base \(e\) logarithms, in conjunction with scientific calculators, is essential for tackling complex mathematical problems.