AS Maths Masterclass

1
AS Maths Masterclass

• Lesson 5
• Exploring logarithms

2
Learning objectives

• The student should be able to
• relate indices to logarithms and solve simple
  indicial equations
• apply the laws of logarithms to simplify
  mathematical expressions
• solve simple logarithmic equations using the laws
  of logarithms

3
Why bother with logs ?

• Thanks to John Napier/ Henry Briggs 1615
• they were used to simplify very large
  calculations in physics and astronomy.

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But are logs actually used nowadays ?

• Finance compound interest
• Photography/ computing image compression
• Computer games measuring algorithm efficiency
• Cinema flicker in a motion picture
• Music pitch, acoustics, sound intensity
• Chemistry ph scale, Newtons law of
  cooling
• Physics radioactive decay, capacitor
  discharge
• Biology/ Economics/ Geog. population growth
• Geology earthquakes measured on the richter
  scale.

5
Recall indices from GCSE ?

• Multiplication rule
• Division rule
• Power rule
• Reciprocal rule
• Power of zero
• Power of a product

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What is a logarithm?

• The log of a number to a given base is the
• index of the power to which the base must
• be raised in order to obtain the original
• number.
• e.g. when
• If then x is called
• the log of N to the base a, written

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Common logs and Natural logs

• If the base is 10 we say that the log is ..
• common. We write log or just lg
• If the base is e (2.718 281 828 ) then we
• say that the log is ..
• natural. We write ln
• Extension activity investigate the number e.

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Evaluating logarithms

• If we write instead of
• then we can easily find logs to any base
• provided the number has a nice
• relationship.
• E.g. Let then
• Hence, x 4 since 3.3.3.3 81

Click here to practice finding logs to any base
(nice numbers) Click here to view the Logarithm
spreadsheet.
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Moving towards the laws of logs

• We all know that
• so by definition,
• In addition, we all know that
• and so again by definition,
• We now need to examine equivalent laws
• for logs like we had for indices

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The first law of logs

• The log of a product is the sum of the logs
• i.e. Prove that
• Let u and v
• Then, by definition we have and
• Hence,
• so by definition, u v
• Finally,

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The power law

• The log of a number to a power is the product of
• the power with the log of the number
• i.e. Prove that
• Let u so and
• Hence,
• so by definition,
• From which,

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Practice with the log laws

• Lets see how these laws work out in practice.
• Teacher clicks here for demonstration of log laws
  with numbers
• Students click here to practice the algebra of
  log laws

Extension activity Click here for harder
questions using the log laws
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Changing the base

• Whenever the numbers are not nice, we can
  always
• change the base to one more convenient.
• Prove
• Let u and so
• Hence,
• so meaning that u
• Finally,

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Practice with change of base

• Click here to practice change of base for
  "awkward logs"l
• Click here for the IWB exercise on logs

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Simple indicial equations

• If, when solving equations we have a variable in
  the power,
• then we say that we have an indicial equation
• We can now solve indicial equations by either
• (a) changing the base or
• (b) taking logs and applying the power law
• Click here for some indicial equations using
  method (b)

16
Harder indicial and log equations

• Three index terms is likely to lead to a
• quadratic equation.
• Click here to look at some quadratic indicial
  equations
• Click here to solve logarithmic equations
• Extension activity Click here to try some harder
  log equations