Logarithms

If a,b and c are real numbers where a = b^c and b >1 , then the power c is called the logarithm of the number a to the base b and c= log_ba

The  exponential function f(x) = a^x  has an inverse function
f^-1(x) = log_ax  upon reflection in the line y = x

If y = a^x , then x = log_ay

 

Example

                 If   y = 5^x
                        x = log₅ y

Examples

 

Some Number base tables

<=  Increasing                  Base 10                   Decreasing =>
Thousands	Hundreds	Tens	Units	tenths
1000	100	10	1	0.1
103	102	101	100	10-1
1000= 103	100= 102	10= 101	1= 100	0.1= 10-1
3 =log101000	2=log10100	1=log1010	0 =log101	-1=log100.1

 

<=  Increasing          Base 2                Decreasing =>
Eights	Fours	Twos	Units	halves
8	4	2	1	1/2
23	22	21	20	2-1
8=23	4=22	2=21	1=20	1/2=2-1
3 =log28	2 =log24	1 =log22	0 =log21	-1=log20.5

 

<=  Increasing                  Base 3                   Decreasing =>
Twentysevens	Nines	Threes	Units	thirds
27	9	3	1	1/3
33	23	31	30	3-1
27=33	9=32	3=31	1=30	1/3 =3-1
3 =log327	2=log39	1=log33	0=log31	-1=log3(1/3)

 

Scientific calculators usually have buttons for :-

Common logarithms :  log  for logarithms to base 10

 natural logarithms  :  ln  for logarithms to base e =2.71828…

These are usually written without a base

log x means log₁₀x  e.g.    log ₁₀45 is written log45
ln x means log_ex   e.g. log _e45 is written ln45 

To convert from any base

Example

Example

Evaluate log₃81

 

If a calculator is allowed :

or

Otherwise,

What power must 3 be raised to make 81 ?

3⁴ = 81 , so x = 4 and log₃81 = 4

   Note :

Example

Log Laws

 

 

Examples

 

Logarithmic equations

Examples