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Logarithms

Maths: Edexcel A Level Pure Maths, topic 6.1.2

Logarithms

In order to solve equations with exponential functions in them, we need to define the inverse of the exponential operation.

Logarithm

Logarithm

  • The logarithmic function is written as log⁡b(x)=y\log_b(x)=y, which is equivalent to x=byx=b^y.
  • bb is called the base of the logarithm.
  • To remember this, think "left to the right equals middle".
  • The base 2 logarithm of the number 8 is 3, so log⁡2(8)=3⇔23=8\log_2(8)=3\Leftrightarrow2^3=8
Range and domain

Range and domain

  • The range of the exponential function is the set of positive real numbers.
  • This means the domain of a logarithm is the set of positive real numbers.
  • So we can only define log⁡b\log_b for x>0x>0.
  • We also specified that the allowed range of values of the base bb of the exponential function was the positive real numbers excluding the number 1.
  • So we can only define log⁡bx\log_bx for b>0,b≠1b>0,b\ne1.
Properties

Properties

  • For single variables and numbers, we often drop the parenthesis to make logarithms easy to write.
  • The logarithm of the number one is equal to zero for any base, so:
  • log⁡b1=0\log_b1=0
  • For logarithms of base 10, we often drop the base number and write:
  • log⁡10x=log⁡x\log_{10}x=\log x
Graph

Graph

  • The graph of y=log⁡bxy=\log_bx does not cross the yy-axis.
  • The x−x--intercept is 1.
  • The orientation of the graph is different for values of b>0b>0 and 0<b<10<b<1 as shown.

Test yourself on Logarithms

  • What is log⁡1010000=4\log_{10}10000 = 4 in exponential form?

  • What is 10−2=110010^{-2} = \frac{1}{100} in logarithm form?

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