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Chapter 5: Problem 95

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

Short Answer

Expert verified
The properties of the natural logarithm function are: \( ln(1) = 0 \), \( ln(e) = 1 \), the interchangeability between \( ln(x) \) and \( e^x \), the adherence to logarithmic identities like \( ln(a*b) = ln(a) + ln(b) \), and its derivative and integral values in calculus.

Step by step solution

01

Property 1 - Logarithm of 1

The natural logarithm of 1 is always 0. This can be demonstrated as follows: \( ln(1) = 0 \). This is because any number raised to the power of zero equals 1.
02

Property 2 - Logarithm of e

The natural logarithm of \( e \) (Euler's Number approx. 2.718) is equal to 1. Therefore, \( ln(e) = 1 \), because \( e \) is the base of the natural logarithm.
03

Property 3 - Interchangeability

The natural logarithm of a number \( x \) and the exponential form \( e^x \) are interchangeable. This means if \( y = ln(x) \), then \( e^y = x \) and vice versa.
04

Property 4 - Logarithmic Identities

The natural logarithm obeys the logarithmic identities. For example, \( ln(a*b) = ln(a) + ln(b) \) and \( ln(a/b) = ln(a) - ln(b) \) and \( ln(a^n) = n*ln(a) \).
05

Property 5 - Derivative and Integral

The derivative of \( ln(x) \) is \( 1/x \), and the integral of \( 1/x \) is \( ln(x) \), provided \( x>0 \). This property is significant in calculus.

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