Logarithm

A logarithm is an exponent; Log_a x is the exponent to which the base a must be raised to obtain x.

The abbreviation log is used for the word logarithm. Read Log_a x as logarithm of x to the base a or the base a logarithm of x.

To remember the location of the base and the exponent in each form, refer to the following :

Logarithmic Form : y = log_a x               Exponential Form : x = a ^y

*where y is the exponent ;a is the base*

Example (conversion between logarithmic and exponential form):
1. 3² = 9 -------> 2 = log₃ 9
2. 5 = log₁₀₀ 100000 -------> 10⁵ = 1000000
3. 5² = 25 ---------> 2 = log₅ 25

Also:
log_a x = log x divided by log a (log x / log a)


Common and Natural Logarithm

Common Logarithm
     Base is always 10 (log)
log 10 or log₁₀ 10

Natural Logarithm
     Base e logarithm (ln)
ln e = 1 (where e is approximately equal to 2.718281828)
example:

e^x-1 = 4

ln e^x-1 = ln 4

(x-1) ln e = ln 4

x ln e - ln e = ln 4

x - 1 = ln 4

x = ln 4 + 1

x = 0.39

Properties of Logarithms

Product Rule:

log_b xy = log_b x + log_b y

Quotient Rule:

log_b x/y = log_b x – log_b y

Power Rule:

log_b x^r = r log_b x

Special Properties

log_b b^x = x

b log_b x = x

log_b 1 = 0