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14.7PureStretch

Working with Natural Logarithms

The natural logarithm, ln⁡\ln, is the logarithm to base ee — the exact inverse of exe^x. Because they undo each other, equations involving ee and ln⁡\ln become straightforward.

25 min Exponentials and Logarithms
Edexcel AS Level Maths: 14.7 Working With Natural LogarithmsCurated by Kepler, not produced by us. Watch it 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
1

ln is log base e

ln⁡x\ln x means log⁡ex\log_e x. As the inverse of exe^x, applying one then the other cancels out completely.

ln⁡(ex)=x,eln⁡x=x\ln(e^{x}) = x, \qquad e^{\ln x} = x
ln⁡\ln and exe^x are inverse operations.
2

Solving e-equations with ln

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

Example 1Solve ex=7e^x = 7.
1Take ln⁡\ln of both sides: ln⁡(ex)=ln⁡7\ln(e^x) = \ln 7.
2x=ln⁡7x = \ln 7.
Answerx=ln⁡7≈1.95x = \ln 7 \approx 1.95
Example 2Solve ln⁡x=2\ln x = 2.
1Exponentiate: eln⁡x=e2e^{\ln x} = e^2.
2x=e2x = e^2.
Answerx=e2≈7.39x = e^{2} \approx 7.39

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

3

Log laws still apply

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

Formula recap

ln⁡x=log⁡ex\ln x = \log_e x
Natural log is base e.
ln⁡(ex)=x, eln⁡x=x\ln(e^{x}) = x, \ e^{\ln x} = x
Inverse operations.
ex=b⇒x=ln⁡be^{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.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Working with Natural Logarithms. Use the feedback to decide what needs another look.

7 past-paper questions · 16 marks

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