The Origin and Discovery of Logarithms
A logarithm, or log, is a mathematical operation. A logarithm consists of a base; when multiplied by itself a specific number of times, it reaches another number. For example, log2(64) equals 6, which means that if you multiply the base 2 six times with itself, it becomes 64. The logarithmic base 2 of 64 is 6. The log base of 10 of 100 equals 2, so you get to 100 by multiplying 10 twice. A logarithm is the inverse of an exponential, that is, 26 equals 64, and 102 equals 100. Sounds complicated to you? John Napier, a Scottish mathematician, discovered logarithms. To further his interest in astronomy, he needed to expedite tedious and complex astronomical calculations involving large numbers. John worked in an age before the invention of calculators.
Logarithms in Real Life
If you have a goal of making a million dollars in the next decade, the formula that your investment advisor will tell you will most likely be a logarithmic one. You will need to invest x at y% annual interest for a decade to get returns worth a million dollars. While it is difficult to visualize a million dollars quickly in your head, notice how much easier it becomes once you know x and y. Logarithms simplify insights involving large figures, such as the number of visits per day on Google’s search home page, earthquake intensity readings, or sound intensity readings of a commercial airplane during take off. There are numerous applications of logarithms due to their ability to “scale down” large numbers in a human-friendly manner. Even after the invention of calculators and supercomputers, centuries after John Napier’s discovery, logarithms are still in use.
Real-life Uses of Logarithms
Think of a scenario where you need to interpret extremely large numbers and dumb them down. You will need logarithmic operations to get consumable insights. How does it simplify computation for a computer? Binary search algorithm design is an example of how logarithmic operations can help locate any name among a million in a phone directory with just 20 comparisons.
Another modern-day application is in space shuttle launches. The velocity, distance traversed, and curved path of a falling rocket incorporate logarithmic operations.
Log transformation of images involves replacing every pixel of an image with its log. The transformed image appears more enhanced with better quality. Logarithmic image processing (LIP) models are mathematical frameworks used extensively in the field of digital image processing algorithms. LIP models improve characteristics such as contrast and sharpness of an image. The applications of logarithmic image transformation extend to various fields, including medical imaging, satellite imaging, robotic vision, and remote sensing.
How Logarithms Make Our Life Easy?
Logarithms rescue us from having to process both large numbers as well as extremely small numbers. For instance, how can you determine how safe it is to indulge in a risky habit such as smoking, driving a bike without a helmet, or bungee jumping down a valley? There is data available in these scenarios that will read something like this hypothetically. In your country, 1 in 20,000 people died in bike accidents because they weren’t wearing a helmet. How do you use this information? Every year, 1 in 100,000 people dies from bungee jumping. Does that mean it is safer than riding a bike without a helmet? The larger the numbers, the more scrambled the math gets in our brains. Logarithms to the rescue! Think of a safety indicator on a scale of 1 to 10, with 1 being the most unsafe and 10 being the safest activity. If 1 in X people succumbs to their indulgence in an activity, the safety measure of that activity will be log10 (X). The log10(20000) value is 4.3, and the log10(100000) value is 5. This can be interpreted to mean bungee jumping is slightly safer than riding a bike without a helmet!
Measuring a substance’s acidity or alkalinity is easier with logarithms too. Our brains find it difficult to process numbers that are too small! For instance, water contains 1*10-7 moles of hydrogen ions per liter. So, is it acidic? And how acidic is it compared to washing soap? This is where a standard pH scale comes in handy and rescues you from having to bother with extremely small numbers. The pH scale ranges anywhere between 0 and 14, making water a neutral solution with a pH of 7. Substances such as washing soap with a pH greater than 7 are acidic.
A logarithm is an instinctive concept by nature. Thanks to this, it has widespread applications in areas that require converting extremely large and extremely small numbers into understandable insights.
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