6.1.4

Natural Logarithm

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Natural Logarithms

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

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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.
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Natural logarithms

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

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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