Surds, Indices, and Logarithms Formulas and Tricks

Surds and Indices

If a and b are non-zero rational numbers and m and n are rational numbers, then

  • a0=1

  • am=1am

  • am=a(1m)

  • am/n=amn

  • (an)n=(a1n)n=a

  • abn=anbn

  • abn=an/bn

  • (an)m=amn

  • nam=amn

  • am×an=am+n

  • am÷an=amn

  • (am)n=amn

  • (abm)n=ambm

  • amn=a(mn)=araisedtothe(mraiswdtothepowern)

  • Ifan,thenm=n

  • Ifam=bmandm0,thena=bifmisoddanda=±bifmiseven

Logarithms

Logarithm: If a is a positive real number, other than 1 and am=x , then we write m=logax and say that the value of log x to the base a is m.

Properties of Logarithms:

  • loga(xy)=logax+logay

  • loga(xy)=logaxlogay

  • logXx=1(i.e.Logofanynumbertoitsownbaseis1)

  • loga1=0(i.e.Logof1toanybaseis0)

  • logbmn=nlogbm

  • logbm=logamlogab=logbm×logba

  • logax=1logXa

  • g. logax=llogbx/logba =logx/loga(Changeofbaserule)

  • When base is not mentioned, it is taken as 10

  • Logarithms to the base 10 are known as common logarithms

  • The logarithm of a number contains two parts, namely characteristic and mantissa. The integral part is known as characteristic and the decimal part is known as mantissa.

    • Case 1: When the number is greater than 1. In this case, the characteristic is one less than the number of digits in the left of decimal point in the given number.

    • Case 2: When the number is less than 1. In this case, the characteristic is one more than the number of zeroes between the decimal point and the first significant digit of the number and it is negative. For example

      Number and Characteristics
      Number and Characteristics

      Number

      Characteristic

      234.56

      2

      23.456

      1

      2.34

      0

      0.234

      -1

      0.0234

      -2

      0.00234

      -3

    • III. For mantissa, we look through the log table.

    • IV. Antilog: If logx=y , then antilog y=x .