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Natural Logarithm

Maths: Edexcel A Level Pure Maths, topic 6.1.4

Natural Logarithms

We often use a specific number as the base of our logarithms.

The number e

The number e

  • The number ee is an important irrational number in mathematics.
  • The number ee can be defined as the base of an exponential function y=exy=e^x whose slope is equal to the function itself.
  • The value of ee is approximately 2.71828.
Natural logarithms

Natural logarithms

  • Logarithms with base ee are called natural logarithms.
  • Natural logarithms have their own notation:
  • log⁡e(x)=ln⁡(x)\log_e(x)=\ln(x)
  • They are defined as the inverse of the exponential function of base ee:
  • ln⁡(x)=y⇔ey=x\ln(x) = y \Leftrightarrow e^y=x
Properties

Properties

  • Natural logarithms follow the same inverse, product, quotient and power rules as other logarithms:
  • ln⁡(2)+ln⁡(3)=ln⁡(6)\ln(2) + \ln(3) = \ln(6)
  • ln⁡(x6)=6ln⁡(x)\ln(x^6)=6\ln(x)

Test yourself on Natural Logarithm

  • What is the correct notation for the natural logarithm of xx?

  • Which of the following expressions is equal to ln⁡7+ln⁡x+ln⁡y\ln7+\ln x+\ln y

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