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Chapter 3: Problem 123

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.

Short Answer

Expert verified
The statement makes sense because the bases 10 and \(e\) are the most practical for calculators, which typically have logarithm functions for these bases. However, mathematically any positive number other than 1 can be used as the base for a logarithm.

Step by step solution

01

Understanding of the Logarithm

A logarithm is the exponent to which a certain base number needs to be raised to obtain another number. In the topic of change-of-base, it's a formula used to convert logarithms from one base to another, with the form \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), where a, b and c are positive numbers, b and c are not equal to 1, and a is not zero.
02

Assessment of the Statement

The statement makes sense because the logarithm base in calculators typically comes in two forms: base 10 (common logarithm) and base \(e\) (natural logarithm). So using base 10 or base e is most practical when using calculators to calculate logarithms, but mathematically it could be any positive number not equal to 1.
03

Reasoning Behind the Practicality of Bases 10 and e

The reasoning behind using base 10 and \(e\) as practical bases comes from their widespread usage in science, engineering, and mathematics. Base 10 is commonly used due to our decimal number system, while base \(e\) has special importance in calculus and exponential growth models.

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