May 17, 2008

Logarithm In Medicine

pH of common materialsMathematics plays a very critical role in all the branches of medicine. Calculus, exponentials and logarithms are no exceptions. They make calculations less cumbersome and easy in many circumstances. Take for example, the plight of little Johnny. He wants to plot the intensity of various sounds that we hear, in a graphical scale. He knows that a barely audible whisper (threshold of hearing) is 10-12 watts/m2 and that of a very loud sound, for example, a jet plane taking off is 1013 times this! Oh, what an exceptional range of hearing this is. Even when Johnny is taking the height of this barely audible sound as 1 mm (above the baseline, in the Y axis), do you think he can manage the jet engines roar in the graph paper? No, not in a linear scale.
Joanna is in a fix too. She fights hard to memorize the number of hydrogen ions present in one liter of pure water (at 25 degree Celsius), in moles per liter. It is 10-7 moles per liter (or a meager 0.0000001 mole in a liter!) She can't even dare to memorize the concentration of hydrogen ions in human arterial blood. Can you help her?

John Napier of Scotland, in 1614, lent a helping hand to the humanity by formulating his famous logarithms. Now Johnny can easily depict the ear's sensitivity in this whole new log scale, or Joanna can remember the number of moles of H+ per liter (concentration) easily, in terms of pH. The graph is squeezed, shrunk-fit into the graph paper now, with the region of interest expanded and highlighted; but extremes not much decompressed.

The log scale can be thought of as the reverse of exponential scale. It is denoted as following: logab=x, means ax=b. Thus log10100=2. In this notation, 'a' is called the 'base', and the equation is pronounced as logarithm of 'b' to the base 'a' is 'x'. Normally, in logarithms, bases used are 10 and 'e'.

Logarithms are used in measuring sound intensity, in a relative scale. If one asks you how much loud is John's voice over Joanna's? You wouldn't say that it was louder by 50 (or 17), would you? Rather you would compare them and are more likely would say that it was double or triple that of Joanna's. Our ears indeed work in a 'multiplying scale' rather than simple 'interval scale' or arithmetic scale. The volume control 'variable resistors' of electronic devices such as transistor radios are logarithmic potentiometers to suit our 'log' loving ears and each successive keys of piano keyboards have frequencies that are multiples of their previous keys (hence the concept of octaves: there are 12 keys and the frequency is increased by twelfth root of 2, in each successive step; thus the frequency of the twelfth pitch following a given pitch will be exactly double = octave). Sound intensity is compared with another sound (threshold of hearing) in a logarithmic scale and the intensity (relative, of course!) of the test sound is given in 'decibel'. The intensities of earthquakes (in Richter scale, as is found by seismograph activities) are also relative ones, an earthquake measuring '5' on Richter scale is 10 times the intensity of '4' (and 1/10th that of '6') in the same scale. Likewise, the hydrogen ion concentration at pH 5 is 100 times than that of pH 7. Situation looks apparently opposite due to the fact that pH is -log (negative logarithm) of hydrogen ion concentration, which is expressed as 10 raised to the power -n. 10 raised to the power -3 is MORE than 10^-5. It must be remembered that though there is zero ('0') in the log scale; you can NOT calculate the logarithm of zero, as 10 raised to the power of anything will NOT give you "0".

Ever looked at your transistor radio's dial? Frequencies of radio stations are plotted in a log scale too. When you want to hear your rock music louder, you turn the volume control knob. This knob is actually a logarithmic potentiometer which allows equal change in loudness for equal rotation, using log rule for the track.

If you slapped a person real hard, he would certainly feel pain! But why should you hurt someone anyway! Weber Fechner law states that the magnitude of pain felt was proportional to the log of intensity of the stimulus (your slap). However, recently power functions are deemed more fit to calculate the magnitude of sensation felt.

In pharmacology too, log-dose response curve is a valuable tool in calculating the concentration of chemicals in bio-assays. During the exponential phase of bacterial cell division, their numbers double after every step. Logarithms can be used to plot the number of cells, in this log phase, in Microbiology. Physiology has plenty of its applications in the Nernst Equation (which describes the potential across the cell membrane for a given ion), Goldmann equation (calculates the potential for all the ions present), Henderson Hasselbalch Equation (governing the relationship between pH and pK) and many many others.

Thus, even in biological systems we can see how logarithms abound and how we can manipulate these concepts into instrumentation or in making bacterial colonies.

Last modified: Mar10,2014
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