Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

Logarithms

Rules of logarithms - log10 and loge for numbers ranging 1 to 1000

Sponsored Links

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

SystemLog to the base ofTerminology
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

 

Numbers - ln, log e and log 10

Sponsored Links

Related Topics

Related Documents

Tag Search

  • en: logarithms
Sponsored Links

Search the Engineering ToolBox

Engineering ToolBox - SketchUp Extension - Online 3D modeling!

3D Engineering ToolBox Extension to SketchUp - add parametric components to your SketchUp model

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the SketchUp Pro Sketchup Extension Warehouse!

Translate this page to
About the Engineering ToolBox!

Citation

This page can be cited as

  • Engineering ToolBox, (2009). Logarithms. [online] Available at: https://www.engineeringtoolbox.com/logarithms-d_1246.html [Accessed Day Mo. Year].

Modify access date.

Customize Ads in the ToolBox

Make ads more useful in Google Ad Settings .

. .

close

3D Engineering ToolBox - draw and model technical applications! 2D Engineering ToolBox - create and share online diagram drawing templates! Engineering ToolBox Apps - mobile online and offline engineering applications!

Scientific Online Calculator

Scientific Calculator

6 22

Sponsored Links
.