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Logarithms

10 min read

Standard form

Standard form writes very large or very small numbers as where . For example and .

Worked exampleStandard form

Write 6 300 000 in standard form.

  1. Place the decimal point after the first digit: 6.3.
  2. Count how many places the point moved: 6.

Answer:

What is a logarithm?

Logarithm
The logarithm of a number is the power you must raise a base to in order to get that number. If then .
Powers and logarithms are opposites. We write log N for base 10.
Exponential and logarithm graphsThe graphs of y equals 2 to the power x and its inverse y equals the logarithm of x to base 2. Each is the mirror image of the other in the line y equals x. The exponential graph passes through (0, 1) and the logarithm graph through (1, 0).xyy = xy = 2ˣy = log₂ x(0, 1)(1, 0)
Try itLogarithm explorer

Slide and read . Doubling the number adds the same amount each time: that is why logs turn multiplying into adding.

Laws of logarithms

LawRule
Productlog(AB) = log A + log B
Quotientlog(A ÷ B) = log A − log B
Powerlog(Aⁿ) = n log A
Special valueslog 1 = 0 and log 10 = 1
Worked exampleUsing the laws

Simplify .

  1. Add logs to multiply.
  2. 1 000 is 10³.

Answer: 3

Check yourself What is log 1000?
Check yourself Write 0.0045 in standard form.

Key terms

Standard form
A × 10ⁿ with 1 ≤ A < 10.
Logarithm
The power to which a base is raised to give a number.
Antilogarithm
The number whose logarithm is given.

Summary

  • Standard form: A × 10ⁿ with A between 1 and 10.
  • log N = x means 10ˣ = N.
  • log(AB) = log A + log B; log(A/B) = log A − log B; log(Aⁿ) = n log A.
Quadratic Equations Congruence