User:Milton Beychok/Sandbox

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.

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 $\log _{b}(xy)=\log _{b}(x)+\log _{b}(y):$

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 $\log _{b}(x/y)=\log _{b}(x)-\log _{b}(y):$

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 963.64^1/3 using $\log _{b}(x^{y})=y\log _{b}(x):$

(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 10^x.

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 ${\mbox{pH}}=-\log _{10}\alpha _{\mathrm {H} ^{+}}$  where $\alpha _{\mathrm {H} ^{+}}$ is the activity of dissolved hydrogen ions in the solution.

• In the field of acoustics, sound pressure level (SPL) or sound level $L_{p}$ is a logarithmic measure of the root mean square (rms) sound pressure of a sound relative to a reference value. It is measured in decibels (dB) and defined as $L_{p}=20\log _{10}(p_{\mathrm {rms} }/p_{\mathrm {ref} })$  where $p_{\mathrm {ref} }$ is the reference sound pressure and $p_{\mathrm {rms} }$ is the rms sound pressure being measured.

• 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 energy released. Each two number increase corresponds to about 1000 times the energy released.

• The Clausius-Clapeyron equation used to characterize the transition between vapor and liquid phases of a substance in the fields of thermodynamics, physical chemistry and chemical engineering involves a logarithmic function. The Antoine equation for determining the vapor pressures of liquids also involves a logarithmic function.

• The log-normal distribution equations used in the field of statistics are logarithmic functions.

• The Fenske equation used for calculating the minimum number of theoretical equilibrium stages required to separate a binary liquid mixture in a continuous distillation column involves logarithms.

• The equation defining the half-life of a radioactive substance involves the logarithm of 2.

• A great many sets of scientific data are correlated by plotting the data in graphs that use one or two axes that are logarithmic (referred to as semi-log or log-log graphs).

• There are quite literally hundreds of other equations used in mathematics, physics, chemistry, engineering, biology, and statistics that involve logarithms.