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Properties of Logarithms

Maths: Edexcel A Level Pure Maths, topic 6.1.3

Properties of Logarithms

We can use the properties of exponents to find ways of combining logarithms of the same base.

Inverse

Inverse

  • The definition of a logarithm as the inverse of an exponent allows us to write the following expressions:
  • log⁡bbx=x\log_bb^x = x
  • blog⁡bx=x,x>0b^{\log_bx} = x, x>0
Product rule

Product rule

  • Adding logarithms of the same base follows the rule:
  • log⁡bx+log⁡by=log⁡bxy\log_bx+\log_by=\log_bxy
Quotient rule

Quotient rule

  • Subtracting logarithms of the same base follows the rule:
  • log⁡bx−log⁡by=log⁡bxy\log_bx-\log_by=\log_b\frac{x}{y}
Power rule

Power rule

  • The logarithm of a number raised to a power can be rewritten as follows:
  • log⁡bxy=ylog⁡bx\log_bx^y=y\log_bx

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