ζC2 – LogarithmsDr J Frost

Starter

Sketch a graph of y = 3x (Note: this was once used as an exam question)

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1 mark: Two of the three criteria.2 marks: All of the three criteria.

(a) Correct shape in left quadrant.(b) Correct shape in right quadrant(c) y-intercept of 1.

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Functions and their inversesSome operators exist to provide the opposite of others.

4 x × 3 x 3 4Function Inverse

4 x + 3 x - 3 4

4 x2 √x 44 x5 5√x 4

12

7

16

1024

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Functions and their inversesSome operators exist to provide the opposite of others.

Function Inverse

4 3x log3 x 481?

How therefore would describe the effect of log3 x in words?

log3 xIt finds the power that, when 3 is raised to it, gives you x.

i.e. If y = log3 x, then 3y = x

It is the opposite/inverse of exponentiation.

We describe this as the “logarithm of x base 3” or “log of x base 3” or “taking the log of x base 3”.

log2 8 = 3

Computing logsRemember that logarithms find the missing power.

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Bro Tip #1: Imagine what power would slot in the middle of the two.

log2 8 = 33= Click to start bromanimation

log3 9 = 2?

log10 100 = 2?

log4 1 = 0?

Bro Tip #2: loga 1 = 0 (a > 0)

log3 3 = 1?

Bro Tip #3: loga a = 1 (a>0)

log2( ) = -112

Computing logsRemember that logarithms find the missing power.

?Bro Tip #4: When we take the log of any value between 0 and 1 (exclusive), we end up with a negative number.

log2( ) = -318

log3( ) = -4181

log4 (-1) =

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__πi__loge 4

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Bro Tip #5: If you want a real result, you can only take logs of positive values.

-4 -2 2 4 6 8 10 12 14

8

6

4

2

-2

-4

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y = log2 xx 0.25 0.5 1 2 4 8

y -2 -1 0 1 2 3? ? ? ? ? ?

Click to brosketch

Using logsLogs help us solve equations when the power is unknown.

Find the x for which 10x = 500

We can write this as x = log10 500

Broculator Tip:The log button on your calculator is implicitly base 10. So “[log] [500]” will give you log10 500

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log9 81 = 2

log2 8 = 3

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23 = 8

92 = 81

Bro Tip: In both cases, the 2 is the ‘base’.

log3 81 = 4?34 = 81

log3 55 = x?3x = 55

More on rewriting Powers as Logarithms

ExercisesC2 Chapter 3 Pg 42

Exercise 3B (All questions)Exercise 3C (Q1, 3, 5, 7)

Laws of LogsThese are 3 laws of logs that you need to remember.

loga xy = loga x + loga y

loga = loga x - loga y

loga (xk ) = k loga x

Proving these involves rewriting the logs as exponential expressions, then using laws of indices.

xy

log3 (a2b)

log2 5log230 – log26

2log3 a + log3 b

log4 ( )

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3log4 (a) – 4log4 b

3loga (a√b)

a3

b4

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Put in the form k + loga (..)

Write the following as a single logarithm.

Laws of Logs

2loga b

2loga b + 3logacloga(b2c3)

4loga(√b)

1 + loga b

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loga(a√b) 12

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Now the other way round! Write in the form loga x, loga y and loga z.

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-loga xloga( )1x ?

Laws of Logs

Using logs in scienceThe well-known “Moore’s Law” states that the processing power of computers doubles every two years. It’s been remarkably accurate so far!

If we were to plot the number of transistors against the year, the type of graph would be exponential.The graph would look rubbish if we chose a range of values on the y-axis to accommodate all the values, because except for the last few years, most of the points would look close to 0.

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Using logs in scienceThe well-known “Moore’s Law” states that the processing power of computers doubles every two years. It’s been remarkably accurate so far!

This graph gets around the problem by letting the y-values increase by a factor of 10 for each unit, rather than increasing by a constant amount each time. Technically this is not allowed!

Using logs in scienceThe well-known “Moore’s Law” states that the processing power of computers doubles every two years. It’s been remarkably accurate so far!

We could instead use a logarithmic scale. We can take the log base 10 of these values. Then we’ll get 3, 4, 5, 6, ..., which is now allowed!*

log 10

(Num

ber o

f tra

nsist

ors)

3

4

5

6

7

8

9

10

Logarithmic scales turn exponential graphs into linear ones (i.e. a straight line), thus making it much easier to plot all the points together.

* Although realistically, a scale of 10, 100, 1000, etc. is permissible as long as we’re mindful that it’s a logarithmic scale.

Using logs in scienceLogarithmic scales are used for earthquakes and noise levels.

From our laws of logs, in base a...

loga ax = 1 + loga x

when a quantity gets a times bigger,the overall result only increases by 1.

Thus using logarithms turns a factor difference into a constant difference.

The Richter Scale is used to measure the magnitude of earthquakes. The scale is logarithmic (base 10): it means if amplitude of the earthquake’s waves gets 10 times bigger, the value on the Richter Scale only increases by 1.

Earthquakes of magnitude 6 vs 7 doesn’t look like a substantial difference, but just the one point difference means it’s ten times worse!

ExercisesC2 Chapter 3 Pg 42

Exercise 3D (All questions)

We saw how we can solve equations like 10x = 125.But what about when the base is different, e.g. 3x = 20?

OPTION 1: The “Look at me, I have a fancy calculator” method

x = log3 20? ?

OPTION 2: The “change of base” method

3x = 20log10 3x = log10 20?

Super Bro Tip:Whenever you’re trying to solve an equation where the variable appears in the power, your first instinct should always be TAKE LOGS DAMMIT!

x log10 3 = log10 20x = log10 20

log10 3

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Solving ax = b

We saw a second ago that we could “change the base” to find log3 in terms of log10.

x = log3 20

3x = 20

x = log10 20log10 3

METHOD 1 METHOD 2

More generally, to change the base from a to b:

loga x =

logb xlogb a

Click to start bromanimationxa

Changing the Base

log10 5log10 2

log2 5 in base 10 ?

Express these logarithms in the specified new base.

log12 10log12 7

log7 10 in base 12

log9 5log9 10

log10 5 in base 9

log10 10log10 5

log5 10 in base 10___1___

log10 5=?

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Broculator Tip:This is how you could find log25 on a calculator if you didn’t have the fancy extra log button. i.e. Change to base 10!

All your base belong to us

Bro Tip: When you switch the argument and base, you take the reciprocal.

loga b = ___1___logb a

All your base belong to us

Variables appear in powers, so apply Bro Tip.(The base of the log doesn’t matter)log 7x+1 = log 3x+2

x = 2log 3 – log 7log 7 – log 3

7x+1 = 3x+2

(x+1)log 7 = (x+2)log 3xlog 7 + log 7 = xlog 3 + 2log 3

xlog 7 - xlog 3 = 2log 3 – log 7x(log 7 - log 3) = 2log 3 – log 7

Solving Equations involving Variables in Powers

Bro Tip:By recognising that 52x = (5x)2, we’ve turned the equation into a quadratic!

(5x)2 + 7(5x) – 30 = 052x + 7(5x) – 30 = 0

Let y = 5x

y2 + 7y – 30 = 0(y+10)(y-3) = 0y = -10 or y = 3

5x = -10 or 5x = 3x = log5(-10) or x = log53

Can’t have log

of a negative

number, so not

a real solution.

Solving Equations involving Variables in Powers

22x + 3(2x) – 4 = 0 3x-1 = 8x+1

x = -3.24x = 0? ?

Solve log2 (2x + 1) – log2 x = 2

When solving, you can often either:a) Get in the form logab = c. Then rearrange as ac = bb) Get in the form logab = logac. Then b = c.

EdExcel exam questions:

Solve log 2 (x + 1) – log 2 x = log 2 7

Exam Technique

Answer: x = 1/2 Answer: x = 1/6? ?

When solving, you can often either:a) Get in the form logab = c. Then rearrange as ac = bb) Get in the form logab = logac. Then b = c.

Edexcel exam questions:

log2 (11 – 6x) = 2 log2 (x – 1) + 3

Exam Technique

Answer: x = 8 or 1/8 Answer: x = -1/4 or 3/2? ?

xx 2

2

22 loglog

16log32log

Solve