Problem 113 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can use any positive number other than 1 in the changeof-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 as one can use any positive number other than 1 in the change-of-base property. However, in practical scenarios, 10 and \(e\) are most commonly used due to their prevalence in calculators and mathematical tools.

Step 1: Understanding the Statement

The first step to solving this problem is understanding the statement. The statement refers to the change-of-base property of logarithms which states that a logarithm base \(b\) can be expressed in terms of any other base \(k\) as long as \(b > 0, b ≠ 1, k > 0\), and \(k ≠ 1\). The statement also mentions the use of calculators, pointing out that most calculators only calculate logarithms with base 10 ('log') and base \(e\) ('ln').

Step 2: Analyzing the Practicality of Bases

The second step involves pondering on the practicality of bases. The reasoning given in the statement that the bases 10 and \(e\) are practical is valid as these are the common logarithms and natural logarithms respectively. These are the ones standard calculators and most mathematical software or tools are set up to calculate.

Step 3: Conclusion

The statement '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' indeed makes sense.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, particularly in algebra and calculus. Put simply, a logarithm asks the question: 'To what power must we raise a certain number (known as the base) to obtain another number?' For example, if we have the equation \( 10^3 = 1000 \), the logarithm of 1000 with base 10 is 3, since \( 10 \), raised to the power of 3, gives 1000. Written in logarithmic form, it reads \( \log_{10}(1000) = 3 \).

Understanding this operation is key to solving exponential equations and growth-decay problems, among others. However, using logarithms isn't always straightforward, as the base can greatly influence the result. Thankfully, we have tools like the 'change-of-base' property which allows us to convert a logarithm to a different base that we may find easier to work with, especially base 10 and base \(e\), due to the availability of calculator functions for these specific bases.

Natural Logarithms

Natural logarithms are a specific type of logarithm where the base is \(e\), an irrational constant approximately equal to 2.71828. The natural logarithm of a number \(x\) is denoted as \(\ln(x)\) and is the power to which \(e\) must be raised to obtain the number \(x\). For instance, if \(e^2 = x\), then \(\ln(x) = 2\).

Natural logarithms are particularly important in mathematics and physics because they appear inherently in continuous growth and decay processes, such as in calculations of interest or the spread of disease. Furthermore, the e-based natural logarithm is the inverse operation of the exponential function \(e^x\), making \(\ln(x)\) a crucial tool in solving equations where the variable is in the exponent of \(e\).

Calculator Logarithm Functions

Calculators have built-in functions to simplify the process of calculating logarithms. These functions are particularly handy because they allow for quick computation without the need to perform the change-of-base manually. Standard calculators typically provide two key logarithm functions:

• The 'log' button which calculates the logarithm of a number with base 10, commonly called the common logarithm.
• The 'ln' button for calculating the natural logarithm with base \(e\), which we discussed earlier.

Using these dedicated buttons, students and professionals can effortlessly solve logarithmic equations and find the values of logarithms during examinations or practical applications, aligning with the established standards within education and various technical fields.