Applications of Logarithms
Logarithms are used in many different ways in real life. Logarithms are often used to make measurements easier to understand. Logarithms can be used to measure things like
Measuring earthquakes on the Richter scale, measuring sounds on the decibel scale, acidic measurement of solutions, plotting big numbers and in data analysis.
Plotting big numbers on the number line in an easily understandable manner.
We know that,
log10 1 = 0
log10 10 = 1
log10 100 = 2
log10 1000 = 3
log10 10000 = 4
log10 10000 = 5,
and so on…
The above logarithmic values can be written exponentially as:
100 = 1
101 = 10
102 = 100
103 = 1000
104 = 10000
105 = 100000
and so on …
Logarithms help us show numbers on scales that are easier to understand. It is easier to talk about a number with 5 digits than it is to say that we have 10,000.
This number line gives the values of log10 1, log10 10, log10 100, …
log10 10000000000.
Richter scale
Seismographs are devices used to record the ground’s movement during an earthquake. They are implanted in the ground throughout the world and operated as part of a global seismographic network.
The seismograph is the output of the seismometer instrument. Typically, this is a graph between ground acceleration and elapsed time at a certain station.
This information allows us to determine the earthquake’s characteristics, such as its duration, peak ground acceleration, kind of waves, and timings, etc.
The magnitude of an earthquake is a numerical representation of the earthquake’s size in terms of energy.
Using seismograph data, the Richter Scale determines the magnitude of every given movement.
The Richter scale is a logarithmic scale with a base of 10. This scale defines earthquake magnitude as the logarithm of the ratio of seismic wave amplitude to an arbitrary standard amplitude.
M = log10 (A ∕ S)
where A is the earthquake’s amplitude, measured with a seismometer from about 100 km away from the earthquake’s epicentre, and S is the standard earthquake amplitude, which is about 1 micrometre.
Decibel scale
Sound transmits energy and its intensity is defined as
I = P/A
Here, P power is the rate at which the particles in the air transfer energy (i.e., KE+PE), and A is the area through which this power is moving.
Sound intensity is measured in decibels, which is measured in terms of a logarithm. Therefore, the intensity of the sound is defined as:
The intensity in decibels = 10×log10 (intensity / intensity of zero decibels)
pH Scale
The term pH is derived from Latin and stands for “potentia hydrogenii” (the power of hydrogen). The pH scale is often used to represent the activity of hydrogen ions.
The pH scale is logarithmic, which effectively means that 1 pH unit represents a 10-fold change.
pH = −log[H+]
A pH of 4 is ten times more acidic than a pH of 5. Likewise, a pH of 4 is one hundred times more acidic than a pH of 6. Similarly, a pH of 10 is ten times more basic than a pH of 9.
Timescales
Timescales of exponential growth or decay can also be calculated using logarithms.
Money grows with a fixed interest rate.
Radioactive decay.
