Natural Logarithms ln(x)

This lesson covers the natural logarithm function, written ln(x), which is the logarithm with base e. Since eˣ is the most important exponential function in mathematics, its inverse ln(x) is the most important logarithm. You need to be confident with ln for differentiation, integration, and solving equations at A-Level.

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Definition

The natural logarithm ln(x) is defined as:

ln(x) = log_e(x)

In words: ln(x) is the power to which you must raise e to get x.

The key relationship is:

eˡⁿ⁽ˣ⁾ = x and ln(eˣ) = x

These express the fact that eˣ and ln(x) are inverse functions of each other.

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The Graph of y = ln(x)

Since y = ln(x) is the inverse of y = eˣ, its graph is the reflection of y = eˣ in the line y = x.