y  ex
Logarithms (and exponentials)
Logarithms are the inverse functions of exponentials, which take the form:
b0
Note:
e.g
y  2x
y   13 
x
y
 5
y  bx
y  3x
yx
y  2x
x
y  log 2 x
b is called the base and x is the exponent. The logarithm to base b
is the power we raise b by to get y. i.e.
logb y  x
e.g
10  1000
2  1024
 log10 1000  3
 log 2 1024  10
3
 2
10
4
 log
y  ln x
4
2
44
The algebraic properties of logarithms can be deduced from their exponential forms
Addition
Powers
Log as a power
x  b A  A  log b x
x  b  A  log b x
x n  b nA  nA  log b x n
y  blogb x
 log b y  log b x
 log b x n  n log b x
 blogb x  x
y  b B  B  log b y
xy  b A B  A  B  log b xy
A
 log b xy  log b x  log b y
Use this to make any
quantity an exponential
using any (positive) base
Subtraction
Converting bases
x  b  A  log b x
x  b A  A  log b x
A
y  b B  B  log b y
x
x
 b A B  A  B  log b  
y
 y
x
 log b    log b x  log b y
 y
log c x  A log c b  A 
Example application
log c x
log c b
log c x
 log b x 
log c b
What are the first twelve
digits of 2100 ?
2100  10log10 2
100
100
2
 10
100log10 2
2100  1030.1029995664
e.g. log10 1000 
ln1000
ln10
2100  100.1029995664 1030
2100  1.26765060023  1030
Note:
ln x  log e x
e  2.71828182846...
Misc
examples
log 42 10 
Natural
logarithm
log x 3  2, x  0
x2  3  x  3
1
ln10 logb 10


log10 42 ln 42 logb 42
42  e3 x
3 x 1  7 4 x  2
ln 42  3 x
ln 42
x
3
  x  1 log 3   4 x  2  log 7
log 3  2 log 7  x  4 log 7  log 3 
log 3  2 log 7
x
4 log 7  log 3
log 3  log 49
x
log 2401  log 3
log 493
x
log 7203
Mathematics topic handout: Algebra - Logarithms (and exponentials) Dr Andrew French. www.eclecticon.info PAGE 1