User:Milton Beychok/Sandbox: Difference between revisions
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Although calculators and computers have essentially replaced the use of logarithms for arithmetic computations (multiplication, division, finding roots, exponentiation, etc.), they are still used for various purposes in many fields. For example: | Although calculators and computers have essentially replaced the use of logarithms for arithmetic computations (multiplication, division, finding roots, exponentiation, etc.), they are still used for various purposes in many fields. For example: | ||
In chemistry, the acidity or alkalinity of a solution is expressed in terms of the [[pH]] scale and pH is defined as | *In chemistry, the acidity or alkalinity of a solution is expressed in terms of the [[pH]] scale and pH is defined as <math>\mbox{pH} = -\log_{10} \alpha_{\mathrm{H}^+}</math> where <math>\alpha_{\mathrm{H}^+}</math> is the [[chemistry (activity)|activity]] of dissolved [[hydrogen ion]]s in the solution.<ref>{{cite book|author=R.Lawn and E. Prichard|title= Measurement of pH (Practical Laboratory Skills Training Guide)|edition=1st Edition|publisher=Royal Society of Chemistry|year=2003|id=ISBN 0-85404-473-6}}</ref> | ||
*In the field of [[acoustics]], [[sound pressure level]] (SPL) or [[sound level]] <math>L_p</math> is a logarithmic measure of the [[root mean square|rms]] sound [[pressure]] of a sound relative to a reference value. It is measured in [[decibel]]s (dB) and defined as <math>L_p = 20 \log_{10}(p_{\mathrm{rms}}/p_{\mathrm{ref}})</math> where <math>p_{\mathrm{ref}}</math> is the reference sound pressure and <math>p_{\mathrm{rms}}</math> is the rms sound pressure being measured.<ref>{{cite book|author=Thomas D. Rossing and Neville H. Fletcher|title=Principles of Vibration and Sound|edition=2nd Edition|publisher=Springer|year=2004|id=ISBN 0-387-40556-9}}</ref> | |||
*In the field of [[seismology]], the magnitude of seismic events such as [[earthquakes]] are measured on a logarithmic scale. Each whole number increase in magnitude represents a tenfold increase in the [[amplitude]] of the eartquake. In terms of [[energy]], each whole number increase corresponds to an increase of about 31.6 times the energe released. Each two number increase corresponds to about 1000 times the energy released.<ref>{{cite book|author=Peter M. Shearer|title=Introduction to Seismology|edition=|publisher= Cambridge Press|year=1999|id=ISBN 0-521-66953-7}}</ref> | |||
In the field of [[seismology]], the magnitude of seismic events such as [[earthquakes]] are measured on a logarithmic scale. Each whole number increase in magnitude represents a tenfold increase in the [[amplitude]] of the eartquake. In terms of [[energy]], each whole number increase corresponds to an increase of about 31.6 times the energe released. Each two number increase corresponds to about 1000 times the energy released.<ref>{{cite book|author=Peter M. Shearer|title=Introduction to Seismology|edition=|publisher= Cambridge Press|year=1999|id=ISBN 0-521-66953-7}}</ref> | |||
Revision as of 18:41, 2 November 2008
The importance of logarithms
The development of electronic calculators and computers in the mid-1900's reduced the importance of logarithms for computations but not the importance of logarithmic functions. Thus, we should discuss the importance of logarithms before and after the advent of electronic calculators and computers.
Before the advent of calculators and computers
The operations of addition and subtraction are much easier to perform than are the operations of multiplication and division. Logarithms were characterized by Pierre-Simon Laplace, the French mathematician and astronomer, as "doubling the life of an astronomer". The German mathematician, Karl Friedrich Gauss, who also did work in physics and astronomy, is said to have memorized a table of logarithms to save the time required to look up a logarithm each time he needed one.[1]
The use of logarithms was widespread because of their relative simplicity compared to multiplication, division, or raising numbers numbers to an exponential power. A few numerical examples, using base-10 logarithms (to eight decimal places) will illustrate that simplicity.
Example 1: Calculate 112.76 × 3,085.31 by using
- log10(112.76) + log10(3085.31 = 2.05215507 + 3.48929881 = 5.54145387
- antilog10(5.54145387) = 347,899.55
The answer would be 347,899.56 by using an electronic calculator.
Example 2: Calculate 47.53 ÷ 860.22 by using
- log10(47.53) − log10(860.22) = 1.67696781 − 2.93460954 = −1.25764172
- antilog10(−1.25764172) = 0.05525233
The answer would be 0.05525233 by using an electronic calculator.
Example 3: Calculate 963.641/3 using
- (1/3) × log10(963.64) = (1/3) × 2.98391482 = 0.99463827
- antilog10 (0.99463827) = 9.87730064
The answer would be 9.87730064 by using an electronic calculator.
Note: The antilog of x is simply the logarithm base raised to the power of x which, in the above examples, is 10x.
After the advent of calculators and computers
Although calculators and computers have essentially replaced the use of logarithms for arithmetic computations (multiplication, division, finding roots, exponentiation, etc.), they are still used for various purposes in many fields. For example:
- In chemistry, the acidity or alkalinity of a solution is expressed in terms of the pH scale and pH is defined as where is the activity of dissolved hydrogen ions in the solution.[2]
- In the field of acoustics, sound pressure level (SPL) or sound level is a logarithmic measure of the rms sound pressure of a sound relative to a reference value. It is measured in decibels (dB) and defined as where is the reference sound pressure and is the rms sound pressure being measured.[3]
- In the field of seismology, the magnitude of seismic events such as earthquakes are measured on a logarithmic scale. Each whole number increase in magnitude represents a tenfold increase in the amplitude of the eartquake. In terms of energy, each whole number increase corresponds to an increase of about 31.6 times the energe released. Each two number increase corresponds to about 1000 times the energy released.[4]
- ↑ R.A. Rosenbaun and G.P. Johnson (1984). Calculus: Basic Concepts and Applications. Cambridge University Press. ISBN 0-521-25012-9.
- ↑ R.Lawn and E. Prichard (2003). Measurement of pH (Practical Laboratory Skills Training Guide), 1st Edition. Royal Society of Chemistry. ISBN 0-85404-473-6.
- ↑ Thomas D. Rossing and Neville H. Fletcher (2004). Principles of Vibration and Sound, 2nd Edition. Springer. ISBN 0-387-40556-9.
- ↑ Peter M. Shearer (1999). Introduction to Seismology. Cambridge Press. ISBN 0-521-66953-7.