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C2: The Laws of Logarithms

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C2: The Laws of Logarithms Learning Objective: to be able to use the laws of logarithms to re-write expressions The Laws of Logarithms: Also, loga (1/x ... – PowerPoint PPT presentation

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Title: C2: The Laws of Logarithms


1
C2 The Laws of Logarithms
  • Learning Objective to be able to use the laws of
    logarithms to re-write expressions

2
Some important results
When studying indices we found the following
important results
  • a1 a

This can be written in logarithmic form as
loga a 1
  • a0 1

This can be written in logarithmic form as
loga 1 0
It is important to remember these results when
manipulating logarithms.
3
The laws of logarithms
The laws of logarithms follow from the laws of
indices
  • The multiplication law

Using the multiplication law for indices
xy am n
Writing this in log form gives
m n loga xy
But m loga x and n loga y so
loga x loga y loga (xy)
4
The laws of logarithms
  • The division law

Using the division law for indices
Writing this in log form gives
But m loga x and n loga y so
5
The laws of logarithms
  • The power law

Using the power law for indices
xn amn
Writing this in log form gives
mn loga xn
But m loga x so
n loga x loga xn
6
The Laws of Logarithms
  • Also, loga (1/x) -loga x

n loga x loga xn
loga x loga y loga (xy)
7
The laws of logarithms
These three laws can be used to combine several
logarithms written to the same base. For example
Express 2loga 3 loga 2 2loga 6 as a single
logarithm.
8
The laws of logarithms
The laws of logarithms can also be used to break
down a single logarithm. For example
Logarithms to the base 10 are usually written as
log or lg.
We can therefore write this expression as
9
Task 1
  • Exercise 3D
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