Calculus How To

Derivative of ln (Natural Log)

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What is a Natural Logarithm?

A natural logarithm (ln) is the inverse function of ex; 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:

ey = x

It’s called the natural logarithm because of the 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:
derivative of lnx

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/x2. 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 (3 min) video to see how the derivative of ln is obtained using implicit differentiation, or read on below:

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

ey = x
.
Step 1: Take the derivative of both sides of ey = x:
derivative ln step 1

Step 2: Rewrite (using algebra) to get:
ln step 2

Step 3: Substitute ln(x) for y:
step 3 ln

References

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

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Stephanie Glen. "Derivative of ln (Natural Log)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/derivatives/derivative-of-ln-natural-log/
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