Logarithmic functions go hand-in-hand with exponential functions. One cannot be explained without the other as both are inverses of each other. When a variable y is the logarithm of another variable x with the base n, represents the relation between the variables x & y as,
y={{\log }_{n}}x \ and \ x={{n}^{y}} \ such \ that \ x>0\And n>0y=log
n
​
x and x=n
y
such that x>0&n>0
The function x={{n}^{y}}x=n
y
is an exponential function and is also the inverse of the logarithmic mapping y={{\log }_{n}}xy=log
n
​
x . The graphs of both these types of functions shown below should make the relation between logarithms and exponents clearer.
The natural logarithm behaves in a similar manner where the base 10 gets replaced by the number e=2.718.., which is an irrational number. The nature of the function graphs still remains the same after the base change. The logarithms used in calculus and scientific models are usually the natural logarithm because the number e and its exponents appear in large number of applications.
Acoustics and noise
Sound is a longitudinal pressure wave that travels through air or other medium as compression and decompression of the molecules of the medium, as shown in the figure below. Different sounds differ in the power and frequencies of their waves and our ear is able to distinguish this difference resulting in us being able to hear a wide variety of different sounds and distinguish between them.
Typically, the human ear is capable of hearing frequencies in the range of 15 Hz to 16000 Hz. This is a wide range of frequencies. Typically, a random noise would consist of not one, but a combination of all such frequencies. To analyze such a wide range of frequencies, use the log scale.
There is a pressure sensor in our ears that senses the pressure changes in the air caused by a sound wave and translates that to a signal, which is then interpreted by the brain. A measure of this pressure created by the sound wave is called the Sound Pressure Level and it is measured in a unit called the decibel scale as,
Sound pressure level, {{L}_{10}}=20{{\log }_{10}}\left( \frac{{{p}_{{}}}}{{{p}_{ref}}} \right)dBL
10
​
=20log
10
​
(
p
ref
​
p
​
​
)dB
Where p is the pressure of the sound wave and p_{ref}p
ref
​
is the reference pressure to compare the pressure of the sound wave with.
Human ear is most sensitive to sounds in the frequency range of around 4000 Hz and normal speech involves frequencies in the range of 100 Hz to 4000 Hz. The noise that we hear is usually a combination of many such frequencies. The resultant sound is the superposition of all sound waves of different frequencies added together to form the combination noise, which we hear. The logarithmic model for sound pressure and power levels describing acoustics provide us a way to perform such analysis.
Binary Search
This is an interesting problem from computer science and one that clearly brings out the effect of logarithms on problems. Given is a dataset of numbers that are already sorted in ascending order, what is the best possible way to search and find whether a number a exists in this dataset?
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