# Logarithms

## Rules of logarithms - log_{10} and log_{e} for numbers ranging 1 to 1000

The logarithm (log) is the inverse operation to exponentiation - and the logarithm of a number is the exponent to which the base - another fixed value - must be raised to produce that number.

The expression

*a ^{y} = x (1)*

can be expressed as the "base *a* logarithm of *x*" as

*log _{a} (x) = y (1b)*

*where *

*a = base *

*x = antilogarithm*

*y = logarithm (log)*

#### Example - Logarithm with base *10*

Since

*10 ^{3} = 1000*

- then the base *10* logarithm of *1000* can be expressed as

*log _{10 }(1000) = 3*

#### Natural Logarithm - Logarithm with base e (2.7182...)

*e ^{y} = x*

*where *

*e = 2.7182.... - e constant or Euler's number *

Base *e* logarithm of *x* can be expressed as

*log _{e }(x) = ln (x) = y*

System | Log to the base of | Terminology |
---|---|---|

log_{a} | a | log to base a |

log_{10} = lg | 10 | common log |

log_{e} = ln | e = 2.718281828459.. | natural log |

log_{2} = lb | 2 | log to base 2 |

### Rules for Logarithmic Calculations

log_{a }(x y) = log_{a }(x) + log_{a }(y) (2)

log_{a }(x / y) = log_{a }(x) - log_{a }(y) (3)

log_{a }(x^{p}) = p log_{a }(x) (4)

log_{a }(1 / x) = - log_{a }(x) (5)

log_{a }(b) = 1 (6)

log_{a }(1) = 0 (7)

log_{a }(0) = undefined (8)

log_{a }(x < 0) = undefined (9)

log_{a }(x) = log_{c }(x) / log_{c }(a) (10)

log_{a }(x → ∞) =(11)∞

#### Example - Logarithm Product Rule

*log _{10 }((5) (6)) = log_{10 }(5) + log_{10}(6)*

* = 0.6990 + 0.7782*

* = 1.4772 *

### Conversion of Logarithms

*lg (x) = lg (e) ln (x) *

* = 0.434294 ln (x) (12)*

*ln (x) = lg (x) / lg (e) *

* = 2.302585 lg (x) (13)*

*lb (x) = 1.442695 ln (x) *

* = 3.321928 lg (x) (15)*

### Log_{10 }(x) and Log_{e }(x) for *x* ranging *1 to 1000*