Logarithms
The 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
Related Topics

Basics
The SIsystem, 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.