Logarithms were invented by the Scottish mathematician and theologian John Napier (1550 - 1617). This was one of the most important contributions to the advance of knowledge. It was through the use of logarithms that Kepler was able to reduce his planetary observations and make his breakthrough discovery of the three laws of planetary motion. This in turn led Newton to develop his law of universal gravitation.
The rules of Napier's original logarithms were unlike those we use today. Henry Briggs (1561 - 1630), working with Napier, developed the more commonly used and convenient base 10 logarithms in which log 1 = 0. Tabulated logarithms enable multiplication and division operations to be done using the much faster operations of addition and subtraction. For example, a short table of logarithms is:
Then, to multiply two numbers, their logarithms are added and the anti-log of the result is the product. Example:N log N 1 0.00000 2 0.30103 3 0.47712 5 0.69897 10 1.00000 20 1.30103 30 1.47712 50 1.69897 100 2.00000 200 2.30103 300 2.47712 500 2.69897
Similarly, for division, logarithms are subtracted and the anti-log of the result is the quotient. Example:20 --> 1.30103 multiplier × 5 --> +0.69897 multiplicand ----------------------- 100 <-- 2.00000 product
300 --> 2.47712 dividend
÷ 100 --> -2.00000 divisor
-----------------------
3 <-- 0.47712 quotient
This principle is used by the slide rule, a computational device now largely replaced by hand-held
calculators.
A basic observable quantity for a star is its brightness. Because stars can have a very broad range of brightness, astronomers commonly introduce a base 10 logarithmic scale called a magnitude scale to classify the brightness of stars. The magnitudes m1 and m2 for two stars are related to their corresponding brightness b1 and b2 by the equation:
The apparent visual magnitudes of some representative objects are shown below:m2 - m1 = 2.5(log b1 - log b2) = 2.5 log (b1/b2)
Faintest naked eye stars +6 approx. Sirius (brightest star) -1.5 Venus (at brightest) -4.4 Full Moon -12.6 Our Sun -26.8
The amount of energy which is transported past a given area of the medium per unit of time is known as the intensity of the sound wave. The greater the amplitude, the greater the rate of energy transport, and the more intense the sound wave is.
The decibel scale defined in terms of the sound wave intensity is given by the relation:
I
L = 10 log ——— [dB]
Io
The parameter I is the intensity of the sound wave and Io is the
reference value. The reference value Io is the threshold of hearing intensity
at 1000 Hz and is 10-12 [W/m2]. The threshold of hearing
has a sound level of 0 decibels (abbreviated 0 dB). Decibels provide a relative
measure of sound intensity. The unit is based on powers of 10 to give a manageable range
of numbers to encompass the wide range of the human hearing response, from the standard threshold
of hearing to the threshold of pain at some ten trillion times that intensity.
The factor of 10 multiplying the logarithm makes it decibels instead of Bels, and is included because about 1 decibel is the just noticeable difference in sound intensity for the normal human ear. Another rule of thumb for loudness is that an intensity of a sound that is 10 times greater than another sounds twice as loud.
Source Power Intensity Threshold of hearing 10-12 W/m2 0 dB Rustling leaves 10-11 W/m2 10 dB Whisper 10-10 W/m2 20 dB Normal conversation 10-6 W/m2 60 dB Busy street traffic 10-5 W/m2 70 dB Vacuum cleaner 10-4 W/m2 80 dB Walkman at maximum output 10-2 W/m2 100 dB Front rows of rock concert 10-1 W/m2 110 dB Threshold of Pain 101 W/m2 130 dB Military jet takeoff 102 W/m2 140 dB
The Richter scale is commonly used to measure the intensity of an earthquake. It is defined as:
2 E
R = --- log ---
3 Eo
Where, Eo = 104.8 Joules is the energy released by a small reference earthquake.
The energy, E can be obtained directly from the Gutenberg-Richter magnitude-energy relation,
here expressed in units of Joules:
Or equivalently:log E = 1.5 × R + 4.8
Since detonation of 1 ton of TNT releases 4.184 × 109 Joules; we can convert these units to equivalent tons of TNT.E = 10(1.5 × R + 4.8)
The Swedish scientist Svante Arrhenius proposed in the late nineteenth century that water dissolves many compounds by separating them into their individual ions. He suggested that acids are compounds that dissolve in water to release hydrogen ions, H+, into solution. He defined bases as substances that dissolve in water to release hydroxide ions, OH-, into solution.
The concentration of ions in solution is commonly abbreviated by using square brackets; that is [H+] means hydrogen ion concentration. Concentration of hydrogen ions [H+] is in units of moles of H+ per liter of solution. In water, the product of [H+] and [OH-] at equilibrium always remains constant:
The parameter KW called the ion-product constant for water. At 25°C, it is equal to 10-14. If the concentration of either H+ or OH- rises, then the other must fall to compensate.KW = [H+][OH-]
In 1909, Sören Sörensen, a Danish biochemist, invented the pH scale for measuring acidity. The pH scale is described by the formula:
The pH scale ranges from 0 to 14. A pH of 7 is considered neutral, below 7 is acidic, and above 7 is basic. Some examples:pH = -log [H+]
[H+] pH [OH-] Example ---------------------------------------------- 100 0 10-14 hydrochloric acid 10-1 1 10-13 stomach acid 10-2 2 10-12 lemon juice 10-3 3 10-11 vinegar 10-4 4 10-10 soda 10-5 5 10-9 rainwater 10-6 6 10-8 milk 10-7 7 10-7 pure water 10-8 8 10-6 egg white 10-9 9 10-5 baking soda 10-10 10 10-4 Tums® (antacid) 10-11 11 10-3 ammonia 10-12 12 10-2 mineral lime 10-13 13 10-1 Drano® (drain cleaner) 10-14 14 100 sodium hydroxide