Transcript Powers and logarithms

Logarithms – making complex calculations easy
John Napier
John Wallis
Johann
Bernoulli
Jost Burgi
Design: D Whitfield, www.pifactory.co.uk
Logarithms
Base
Index
Power
Exponent
Logarithm
2
10
= 100
Number
“10 raised to the power 2 gives 100”
“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
Design: D Whitfield, www.pifactory.co.uk
Logarithms
y = bx
Logby = x
Logarithm
Base
102 = 100
Number
logby = x
is the inverse of
y = bx
Base
Logarithm
Log10100 = 2
Number
23 = 8
Log28 = 3
34 = 81
Log381 = 4
Log525 =2
52 = 25
Log93 = 1/2
91/2 = 3
Design: D Whitfield, www.pifactory.co.uk
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
Design: D Whitfield, www.pifactory.co.uk
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
Design: D Whitfield, www.pifactory.co.uk
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
Design: D Whitfield, www.pifactory.co.uk
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
Design: D Whitfield, www.pifactory.co.uk
103 = 1000
log101000 = 3
p = q2
logqp = 2
24 = 16
log216 = 4
xy = 2
logx2 = y
104 = 10,000
log1010000 = 4
pq = r
logpr = q
32 = 9
log39 = 2
logxy = z
xz = y
42 = 16
log416 = 2
loga5 = b
ab = 5
10-2 = 0.01
log100.01 = -2
logpq = r
pr = q
log464 = 3
43 = 64
c = logab
b = ac
log327 = 3
33 = 27
log366 = 1/2
361/2 = 6
log121= 0
120 = 1
Design: D Whitfield, www.pifactory.co.uk
Laws of logarithms
Every 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 logarithms
Every 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 logarithms
Every 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 logarithms
Every number can be expressed in exponential
form… every number can be expressed as a log
am.an = am+n
loga(xy) = logax + logay
am/an = am-n
loga(x/y) = logax - logay
(am)n
=
am.n
logaxn = nlogax
Design: D Whitfield, www.pifactory.co.uk
Change of base property
Logbx
Logax =
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
Design: D Whitfield, www.pifactory.co.uk