14.7PureStretch

Working with Natural Logarithms

The natural logarithm, $ln$ , is the logarithm to base $e$ — the exact inverse of $e^{x}$ . Because they undo each other, equations involving $e$ and $ln$ become straightforward.

25 min Video by Zeeshan Zamurred Exponentials and Logarithms

Edexcel AS Level Maths: 14.7 Working With Natural LogarithmsWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand ln as log base e
Use ln and eˣ as inverse operations
Solve equations involving e and ln
Apply the log laws to natural logs

ln is log base e

$ln x$ means $lo g_{e} x$ . As the inverse of $e^{x}$ , applying one then the other cancels out completely.

ln (e^{x}) = x, e^{l n x} = x

ln

and

e^{x}

are inverse operations.

Solving e-equations with ln

To solve an equation with $e$ , take $ln$ of both sides; to solve one with $ln$ , exponentiate (raise $e$ to both sides). The inverse relationship clears the other function.

1Take

ln

of both sides:

ln (e^{x}) = ln 7

x = ln 7

Answer

x = ln 7 \approx 1.95

Tip — e and ln are a matched pair: use ln to undo e, and e to undo ln.

Log laws still apply

All the log laws (product, quotient, power) work for $ln$ , since it is just a logarithm with base $e$ .

Formula recap

ln x = lo g_{e} x

Natural log is base e.

ln (e^{x}) = x, e^{l n x} = x

Inverse operations.

e^{x} = b \Rightarrow x = ln b

Solving e-equations.

Common mistakes to avoid

Treating ln and log₁₀ as identical.

ln is base e specifically (≈2.718), not base 10.

Forgetting that e^(ln x) = x.

They cancel — that is the whole point of using ln on e-equations.

Key takeaways

ln x = log base e.
ln and eˣ are inverses: ln(eˣ) = x and e^(ln x) = x.
Use ln to solve e-equations and e to solve ln-equations.

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