Summary Logarithms, Advanced Programme Mathematics - Grade 12 (IEB)

Logarithms
ax = b or logab = x
baseexponent = value logbase value = exponent
base > 0
base ≠ 1
value > 0

Laws:
1. logmxa = alogmx (x > 0 ; y > 0 ; 0 < a < 1 or a > 1)

2. logm(xy) = logmx + logmy (x > 0 ; y > 0 ; 0 < a < 1 or a > 1)

𝑥
3. logm( ) = logmx - logmy (x > 0 ; y > 0 ; 0 < a < 1 or a > 1)
𝑦

𝑙𝑜𝑔𝑛 𝑥
4. logmx = (change of base law)
𝑙𝑜𝑔𝑛 𝑚

1
5. logxm =
𝑙𝑜𝑔𝑚 𝑥



Note:
loga(x + y) ≠ logax + logay
loga(x - y) ≠ logax - logay
. (logax)m ≠ mlogax


Common logarithm
= a logarithm to base 10
(Written without showing the base)
Example: log10x = logx



Deductions:
1. logaa = 1

2. loga1 = 0

3. log1/ax = - logax

1
4. log1/a( ) = logax
𝑥

, Log Expressions
Terms with different bases – simplify the terms separately
Terms with the same base – write as a single log

Example: Law 1,3 and Deduction 1

log927 + log93
= log9(27 x 3)
= log981
= log9(9)2
= 2log99
=2x1
=2

Example: Law 2,3 and Deduction 1

log327 + log33
27
= log3( 3 )
= log39
= log3(3)2
= 2log33
=2x1
=2

Example: Law 1,2,3 and Deduction 1,2

log20 + log 30 - log6 + log41
= log20 + log 30 - log6 + 0
= log(20 x 30) - log6
= log(600) - log6
600
= log( 6
)
= log100
= log10100
= log10(10)2
= 2log1010
=2x1

Example:

log3√27
= log3271/2
1
= 2 log327
1
= 2 log3(3)3
3
= 2 log3(3)
3
= x1
2
3
= 2