Design: D Whitfield, www.pifactory.co.uk

Logarithms – making complex calculations easyJohn Napier

John Wallis

Jost Burgi

Johann Bernoulli

Design: D Whitfield, www.pifactory.co.uk

Logarithms

102 = 100“10 raised to the power 2 gives 100”

Base

IndexPower

ExponentLogarithm

“The power to which the base 10 must be raised to give 100 is 2”

“The logarithm to the base 10 of 100 is 2”

Log10100 = 2

Number

Design: D Whitfield, www.pifactory.co.uk

Logarithms

102 = 100Base

Logarithm

Log10100 = 2

Number

Logarithm

Nu

mb

er

Base

y = bx

Logby = x

23 = 8 Log28 = 3

34 = 81 Log381 = 4

Log525 =2 52 = 25

Log93 = 1/2 91/2 = 3

logby = xis the inverse of

y = bx

Design: D Whitfield, www.pifactory.co.uk

103 = 1000 log101000 = 3

24 = 16 log216 = 4

104 = 10,000 log1010000 = 4

32 = 9 log39 = 2

42 = 16 log416 = 2

10-2 = 0.01 log100.01 = -2

log464 = 3 43 = 64

log327 = 3 33 = 27

log366 = 1/2 361/2 = 6

log121= 0 120 = 1

p = q2 logqp = 2

xy = 2 logx2 = y

pq = r logpr = q

logxy = z xz = y

loga5 = b ab = 5

logpq = r pr = q

c = logab b = ac

Design: D Whitfield, www.pifactory.co.uk

103 = 1000 log101000 = 3

24 = 16 log216 = 4

104 = 10,000 log1010000 = 4

32 = 9 log39 = 2

42 = 16 log416 = 2

10-2 = 0.01 log100.01 = -2

log464 = 3 43 = 64

log327 = 3 33 = 27

log366 = 1/2 361/2 = 6

log121= 0 120 = 1

p = q2 logqp = 2

xy = 2 logx2 = y

pq = r logpr = q

logxy = z xz = y

loga5 = b ab = 5

logpq = r pr = q

c = logab b = ac

Design: D Whitfield, www.pifactory.co.uk

103 = 1000 log101000 = 3

24 = 16 log216 = 4

104 = 10,000 log1010000 = 4

32 = 9 log39 = 2

42 = 16 log416 = 2

10-2 = 0.01 log100.01 = -2

log464 = 3 43 = 64

log327 = 3 33 = 27

log366 = 1/2 361/2 = 6

log121= 0 120 = 1

p = q2 logqp = 2

xy = 2 logx2 = y

pq = r logpr = q

logxy = z xz = y

loga5 = b ab = 5

logpq = r pr = q

c = logab b = ac

Design: D Whitfield, www.pifactory.co.uk

103 = 1000 log101000 = 3

24 = 16 log216 = 4

104 = 10,000 log1010000 = 4

32 = 9 log39 = 2

42 = 16 log416 = 2

10-2 = 0.01 log100.01 = -2

log464 = 3 43 = 64

log327 = 3 33 = 27

log366 = 1/2 361/2 = 6

log121= 0 120 = 1

p = q2 logqp = 2

xy = 2 logx2 = y

pq = r logpr = q

logxy = z xz = y

loga5 = b ab = 5

logpq = r pr = q

c = logab b = ac

Design: D Whitfield, www.pifactory.co.uk

103 = 1000 log101000 = 3

24 = 16 log216 = 4

104 = 10,000 log1010000 = 4

32 = 9 log39 = 2

42 = 16 log416 = 2

10-2 = 0.01 log100.01 = -2

log464 = 3 43 = 64

log327 = 3 33 = 27

log366 = 1/2 361/2 = 6

log121= 0 120 = 1

p = q2 logqp = 2

xy = 2 logx2 = y

pq = r logpr = q

logxy = z xz = y

loga5 = b ab = 5

logpq = r pr = q

c = logab b = ac

Design: D Whitfield, www.pifactory.co.uk

Laws of logarithmsEvery number can be expressed in exponential form… every number can be expressed as a log

Let p = logax and q = logay

So x = ap and y = aq

xy = ap+q

p + q = loga(xy)

p + q = logax + logay = loga(xy)

loga(xy) = logax + logay

Design: D Whitfield, www.pifactory.co.uk

Laws of logarithmsEvery number can be expressed in exponential form… every number can be expressed as a log

Let p = logax and q = logay

So x = ap and y = aq

xy = ap-q

p - q = loga(x/y)

p - q = logax - logay = loga(x/y)

loga(x/y) = logax - logay

Design: D Whitfield, www.pifactory.co.uk

Laws of logarithmsEvery number can be expressed in exponential form… every number can be expressed as a log

Let p = logax and q = logax

So x = ap and x = aq

x2 = ap+q

p + q = loga(x2)

p + q = logax + logax = loga(x2)

logaxn = nlogax

Design: D Whitfield, www.pifactory.co.uk

Laws of logarithmsEvery number can be expressed in exponential form… every number can be expressed as a log

loga(x/y) = logax - logay

loga(xy) = logax + logay

logaxn = nlogax

am.an = am+n

am/an = am-n

(am)n = am.n

Design: D Whitfield, www.pifactory.co.uk

Change of base property

Logax =Logbx

Logba

Design: D Whitfield, www.pifactory.co.uk

Solving equations of the form ax = b

3x = 9

4x = 64

5x = 67

Solve by taking logs:

log5x = log67

xlog5 = log67