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Chapter 1: Problem 49

In your own words, state the properties of the natural logarithmic function.

Short Answer

Expert verified
The natural logarithm has five key properties: its definition as the logarithm to the base \( e \), transformation rules, product rule, quotient rule, and power rule.

Step by step solution

01

Definition of Natural Logarithm

The natural logarithm, represented as \( ln(x) \), is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.718281828.
02

Transformation Rules

The transformation of logarithms include: \( ln(a) + ln(b) = ln(ab) \) (logarithm of the product is the sum of the logarithms) and \( ln(a) - ln(b) = ln(a/b) \) (logarithm of the quotient is the difference of the logarithms).
03

Product Rule

The product rule of logarithms states that the logarithm of a product is the sum of the logarithms of its factors. In other words, \( ln(xy) = ln(x) + ln(y) \).
04

Quotient Rule

The quotient rule of logarithms states that the logarithm of the ratio of two quantities is the difference of the logarithms of those quantities. More specifically, \( ln(x/y) = ln(x) - ln(y) \).
05

Power Rule

The power rule of logarithms says that the logarithm of a number raised to a power is the product of that power and the logarithm of the number. Formally, \( ln(x^n) = n \cdot ln(x) \).

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