Rules of Logarithms

Logarithm is the mathematical function which is used to find the power to raise the base in order to get x. Basic rule for logarithms are how many times number  we are multiplying (base) with how many times to use it in a multiplication (logarithm).

Definition of logarithm
When “b “is raised to power of v and is equal to u.
bv = u
log b (u) = v
Then the base “b” logarithm of “u” is equal to” v”.

Example:
24 = 16
2x2x2x2 = 16
2 = base
4 = logarithm
Log2 (16) = 4

Rules of logarithms
Logarithm rules are useful in many fields such as maths, science, and engineering. Logarithms are used when numbers are very large or very small and are often easier to appreciate their size in this way.
1. Logarithm product rule
log b (uv) =  log b u + log b v
2. Logarithm quotient rule
log b (u/v) = log b (u) – log b (v)
3. Logarithm power rule
log b (u v) = v. log b (u)
4. Logarithm base switch rule
log b (c ) = 1 / ( log c (b)
5. Logarithm base change rule
log b (u) = log c (u) / log c (b)
6. Derivatives of logarithm
F (u) = log b (u) => f’ (u) = 1/ ( u In (b))
7. Integral of logarithm
 ∫ log b (u) du = u. (log b (u) – 1 / In (b)) + c
8. Logarithm of negative number
log b (u) is undefined when u = 0
9. Logarithm of 0
Log b (0) is undefined
    Lim
u -----> 0+ log b (u) = - a
10. Logarithm of 1
log b (1) = 0
11. Logarithm of base
log b (b) =1
12. Logarithm of infinity
lim log b (a) = a when u ------>  a
Where,
b - Positive (b =! 1)
u, v – Positive real numbers

Change of Base
The base of “b” logarithm of “u” is base “c” logarithm of “u” divided by the base “c” logarithm of “b”
 For example log b (u) can be converted by change of base of c.
log b (u) = log c (u) / log c (b)          ( b =! 1, c =! 1) ( u > 0)
A formula which allows rewriting a logarithm in terms of logs written with another base (change of base). This is especially used to evaluate a log to any base other than “10” or “e”, when used in a calculator.
Example:
Log 16 32 = log 2 32 / log 2 16 = 5/4 ˜ 1.33
Power Rule
The logarithm of “u” raised to the power of “v” is “v” times the logarithm of “u”.
log b u v = v (log b u)
Example:
Log 3 9 2 = 2 (log3 9)