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Indices and logarithms

See also: mathematical-content-knowledge, teaching-mathematics

Motivation#

Indicies provide an easy way to write repeated multiplication.

A logarithm is another name for an index. It answers questions like "What power of 2 gives 16?"

Uses of logarithms include

  • decibels
  • the Richter scale
  • pH value in chemistry

When two measured quantities are related by a power law, the parameters can be estimated using log plots - useful in experimental science

Declarative knowledge#

Indices#

A power is a product of equal factors (known as the base). The number of factors is the index or exponent.

Index laws

  • \(a^m \times a^n = a^{m+n}\)
  • \(\dfrac{a^m}{a^n} = a^{m-n}, (provided m>n)\)
  • \((a^m)^n = a^{mn}\)
  • \((ab)^n = a^nb^n\)

Logarithms#

Taking a logarithm is the inverse of taking a power.

Since \(2^3 = 8\), then \(\log_2 8 = 3\)

To find the logarithm of a number \(a\) to the base \(b\), ask "what power do I raise \(b\) to, in order to obtain \(a\)?"

The relationship between logarithms and powers is: \(x = \log_a y\) if and only if \(y = a^x\)

Logarithm laws#

Support \(a > 0\)

  1. \(\log_a 1 = 0\) and \(\log_a a = 1\)

    • log base \(a\) of nothing is \(0\)
    • log base \(a\) of \(a\) is \(1\)

  2. If \(x\) and \(y\) are positive numbers, then \(\log_a xy = \log_a x + \log_a y\)

    • log of a product is the sum of the logs

  3. If \(x\) and \(y\) are positive numbers, then \(\log_a \dfrac{x}{y} = \log_a x - \log_a y\)

    • log of a quotient is the difference of the logs

  4. If \(x\) is a positive number and \(n\) is any number, then \(\log_a \dfrac{1}{x} = -\log_a x\)

    • log of a reciprocal is the negative of the log

  5. If \(x\) is a positive number and \(n\) is any rational number, then \(\log_a x^n = n\log_a x\)

    • log of a power is the power times the log

Procedural knowledge#

Resources#