# Logarithms

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 *

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