Logarithms
The rules of logarithms - log10 and loge 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
ay = x (1)
can be expressed as the "base a logarithm of x" as
loga (x) = y (1b)
where
a = base
x = antilogarithm
y = logarithm (log)
Example - Logarithm with base 10
Since
103 = 1000
- then the base 10 logarithm of 1000 can be expressed as
log10 (1000) = 3
Natural Logarithm - Logarithm with base e (2.7182...)
ey = x
where
e = 2.7182.... - e constant or Euler's number
Base e logarithm of x can be expressed as
loge (x) = ln (x) = y
| System | Log to the base of | Terminology |
|---|---|---|
| loga | a | log to base a |
| log10 = lg | 10 | common log |
| loge = ln | e = 2.718281828459.. | natural log |
| log2 = lb | 2 | log to base 2 |
Rules for Logarithmic Calculations
loga (x y) = loga (x) + loga (y) (2)
loga (x / y) = loga (x) - loga (y) (3)
loga (xp) = p loga (x) (4)
loga (1 / x) = - loga (x) (5)
loga (b) = 1 (6)
loga (1) = 0 (7)
loga (0) = undefined (8)
loga (x < 0) = undefined (9)
loga (x) = logc (x) / logc (a) (10)
loga (x → ∞) = ∞ (11)
Example - Logarithm Product Rule
log10 ((5) (6)) = log10 (5) + log10(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)
Log10 (x) and Loge (x) for x ranging 1 to 1000
Related Topics
-
Basics
The SI-system, unit converters, physical constants, drawing scales and more. -
Mathematics
Mathematical rules and laws - numbers, areas, volumes, exponents, trigonometric functions and more.
Related Documents
-
Algebraic Expressions
Principal algebraic expressions formulas. -
Amines, Diamines and Cyclic Organic Nitrogen Compounds - pKa Values
Values for the negative logarithm of the acid dissociation constant, pKa, of the conjugated acid of amines, diamines and cyclic organic nitrogen compounds, shown together with the molecular structure of the acids. -
Arithmetic and Logarithmic Mean Temperature Difference
Arithmetic Mean Temperature Difference in Heat Exchangers - AMTD - and Logarithmic Mean Temperature Difference - LMTD - formulas with examples - Online Mean Temperature Calculator. -
Constants of e
The mathematical e constants. -
Constants of PI
The mathematical π constants. -
Decibel
Logarithmic unit used to describe ratios of signal levels - like power or intensity - to a reference level. -
Exponents - Powers and Roots
The laws of fractional and integer exponents. -
Factorials
The product of all positive integers. -
Fractions
Law of fractions -
Inorganic Acids and Bases - pKa Values
Values for the negative logarithm of the acid dissociation constant, pKa, of inorganic acids and bases, as well as hydrated metal ions. -
Laws of Indices
Simplifying calculations by involving indices. -
Numerical Constants
Some numerical constants. -
pH - Basic (alkaline) vs. Acidic
Introduction to pH - the acidic and basic (alkaline) definition. -
Phenols, Alcohols and Carboxylic Acids - pKa Values
For oxygen containing organic compounds this is given: pKa (the negative logarithm of the acid dissociation constant), molecular structures, molar weights, density and melting and boiling points. -
Scientific and Engineering Terms - ANSI Abbreviations
Abbreviations for Use on Drawings and in Text - ANSI Y1.1. -
Signals - Adding Decibels
The logarithmic decibel scale is convenient when adding signal values like sound power, pressure and others from two or more sources. -
Solving Quadratic Equation with One Unknown
How to solve a quadratic equation. -
Sound Pressure
Sound Pressure is the force of sound on a surface perpendicular to the propagation of sound. -
Sound Propagation - the Inverse Square Law
Doubling of the distance from a noise source reduces the sound pressure level with 6 decibel. -
Trigonometric Functions
Sine, cosine and tangent - the natural trigonometric functions.
