• MathsFirst Home
• Online Maths Help
  • Notation
  • Arithmetic
    • Decimals
    • Fractions
    • HCF and LCM
    • Order of Operations
    • Signed Numbers
  • Algebra
    • Exponents
      • Integer Exponents
      • Fractional Exponents
      • Exponents Worksheet
    • Combining Like Terms
    • Simple Expansions
    • Factorisation
    • Mathematical Formulae
      • Constructing Formulae
      • Rearranging Formulae
        • Order of Operations for Algebraic Expressions
        • Inverse Operations and Functions
        • Rearranging Equations I (Simple Equations)
        • Rearranging Equations II (Quadratic Equations)
        • Rearranging Equations III (Harder Examples)
      • Area, Surface Area and Volume Formulae
    • Coordinate Systems and Graphs
    • Linear Equations & Graphs
    • Simultaneous Linear Equations
    • Polynomials
    • Quadratic Polynomials
    • Rational Functions
    • Logarithms
  • Calculus
    • Differentiation
      • Tangents, Derivatives and Differentiation
      • The Basic Rules
      • Products and Quotients
      • Chain Rule
      • Logarithmic Differentiation
      • Mixed Differentiation Problems
      • Sign of the Derivative
      • Sign of the Second Derivative
    • Integration
      • Basic Integrals
      • Multiple, Sum and Difference Rules
      • Linear Substitution
      • Simpler Integration by Substitution
      • Harder Integration by Substitution
      • Trig Substitution 1
      • Trig Substitution 2
      • Integration by Parts
  • Trigonometry
    • Pythagoras Theorem
    • Trig Functions and Related Topics
• First Year Maths Pathways
• Readiness Quizzes
• Study Maths
  at Massey
  • Undergraduate
  • Postgraduate
• Other Math Links
• About MathsFirst
• Contact Us

Logarithms

The Natural Logarithm

The common logarithm is used in many scientific applications, such as the Richter scale. In mathematics, the most commonly used base for logarithms is the number e = 2.718281828...

log_e is abbreviated to ln (read "lin") though Matlab uses log for this function.

Thus, by definition y = e^x    ln y = x

Replacing y in the second equation by e^x we get  

Replacing x in the first equation by ln x we obtain

It is common to write e^x = exp(x).

The functions ln and exp are inverses of one another. That is, they cancel each other out.

Note   e^x > 0   for any x.
Consequently,  ln x  is defined only for x>0.

Here are the graphs of ln and exp:

Each graph is a reflection of the other with respect to the line y = x.

<< The Common Logarithm | Logarithms Index | Applications >>