**Contents:**

## What is a Natural Logarithm?

A **natural logarithm** (ln) is the inverse function of e^{x}; It is a logarithm with base e (the base is always a positive number). In other words, y = ln x is the same thing as:

**e**

^{y}= xThis fact comes into play when we’re finding the derivative of the natural log.

It’s called the *natural* logarithm because of the “e” (Euler’s number). Mercator (1668) first used the term “natural” (in the Latin form *log naturalis*) for any logarithm to base e (as cited in O’Connore & Robertson, 2001).

## What is the Derivative of ln?

The derivative of ln(x) or ln(kx) is 1/x. In notation, that’s:

The natural log function, and its derivative, is defined on the domain x > 0.

The derivative of ln(k), where k is any constant, is zero.

The second derivative of ln(x) is -1/x^{2}. This can be derived with the power rule, because 1/x can be rewritten as x^{-1}, allowing you to use the rule.

## Derivative of ln: Steps

Watch this short (2 min) video to see how the derivative of ln is obtained using implicit differentiation. The video also shows how to calculate the derivative of ln(*k*x) and x^{2}:

Can’t see the video? Click here.

To find the derivative of ln(x), use the fact that y = ln x can be rewritten as

**e**

^{y}= xStep 1: Take the derivative of both sides of e

^{y}= x:

Step 2: Rewrite (using algebra) to get:

Step 3: Substitute ln(x) for y:

## References

Exponential Review. Retrieved November 12, 2021 from: https://mste.illinois.edu/malcz/ExpFit/REVIEW-answers.html

Adler, F. (2013). Modeling the Dynamics of Life: Calculus and Probability for Life Scientists. Cengage Learning.

Daugherty, Z. (2011). Derivatives of Exponential and Logarithm Functions.

O’Connor, J. & Robertson, E. (2001). The Number e. Retrieved August 20, 2020 from: https://mathshistory.st-andrews.ac.uk/HistTopics/e/#:~:text=In%201668%20Nicolaus%20Mercator%20published,for%20logarithms%20to%20base%20e.

Ping, X. (2016). Why natural constant “e” is called “natural”.