Integrated Maths 10+

Journal Entry 10: Logarithms before calculators

Currently, logarithm is the unit we are working on in class. A logarithm is basically the power to which the base is raised. So for example; xa=b is equivalent to logxb=a in where x will be the base number. Since it is the 21st century and technology is developing, we have our handy calculators to solve various complex equations and log numbers efficiently. But one would question it’s history and how people worked with log back when calculators did not exist.

Around 400 years ago, logarithms were invented so that large numbers could be multiplied and divided in a faster process (Life without a, 2010). As John Napier, a Scottish mathematician, became tired of solving the complex calculations, he found logarithm which helped speed the process of solving lengthly equations. He also stated that this discovery was to help him in the multiplication of numbers, now known as sines (Murray, 2012). This sine is the value of the side of a righ-angled triangle. Officially he introduced logarithms in 1614 where he published the first tables, as well (Natural logarithm, n.d.). Napier observed the various actions of the points on a straight line for the  ‘Naperian/natural logarithm’. On the straight line the L point for log move from minus infinity to plus infinity and the X point for sine move from zero to infinity at a speed that corresponds to it’s distance from zero (Murray, 2012). Thus, this means that whenever the value of L is zero, X is one and have equivalent speed at that point. This discovery enabled him to see the relationship between arithmetic and geometric cycle, as it supports a rule of multiplication and rasing to a power of the value of the X point corresponds to addition and multiplication of the values of the L point. And later, another mathematician named Henry Briggs followed up with Napier’s research and together they found the system of logarighms with base 10 (Stern, 2007). The change and clashes between both ideas introduced the Briggsian logarithm (common logarithm). Briggs then published tables of logarithms to 14 decimal figures.

These tables, also known as “Log Tables”, were the essential tools people used back then to solve the log calculations without calculators, right after these mathematicians successfully introduced these equipments. On these log tables, there is a list of ‘common logarithms’ which have a base 10. Simply finding the log value (between 1-10) on the table enables you to find it’s converted number. After the tables were published for 10-second intervales in the 18th century, it became very convenient for longer decimals. In order to achieve this, many researchers had to produce a table with finer intervales to be able to calculate small values of logarithmic functions. Although the tables were an important tool, the rules of logarithms helped speed and ease the calculations. These rules include: multiplied/divided can be changed to added/subtracted logs and exponents/roots can be changed to mutiplying/dividing values. The rule for the transformation from multiplication to addition was actually founded during the times the invention of logarithms was forshadowed according to the arithmetic and geometric scales (Murray, 2012). In a geometric sequence the values have a common ratio of 10 and an arithmetic sequence have values with a common difference of 1. When a few geometric values are multiplyed, it equals to the addition of it’s equivalent exponents of the common ratio. With this information, the people have concluded that multiplication can be converted into addition.

Now, using only the rules of log and the log table, I have tried to solve a complicated equation I have made myself. It is both a good experience to observe how people in the past solved these lengthy calculations without calculators and become more aware of logarithm. I have outlined the explanation on the right and inserted my calculations on the left.

………   

After solving the equation, I have checked my answer with the calculator to see how close the answers are to each other.

Ultimately, the answer to the equation was somewhat similar to each other, yet not accurately the same. Although we can tell that people were able to arrive to quiet an reliable answer back then, without a calculator. I think that if the number of decimal places in the tables could increase the accuracy of the calculation. And so if I have used a log table with more decimal places it would have allowed me to reach a more accurate solution, yet the one I have used was only until four decimal places. If the table was more accurate and efficient, the answer would be closer to the solution found on the calculator. Also another issue with accuracy may be from human error. Because human’s are not perfect, like machines, we tend to make simple mistakes we did not even mean to make. Like throughout the process of solving this equation, I have made several mistakes that have eventually caused the whole solution to be way off. Although we are not worried about this because we have our perfect calculators, the people back then may have struggled with this because they did not have any technology. The people may not have been affect too much by the accuracy, unless they were solving a very specific idea in the natural sciences or mathematics. But like us now, we would round the answers throughout and reach a solution that is somewhat similar to the calculator answer, not affecting us very greatly. So, the maximum accuracy possible with manual calculations actually depends on how much finer the values are in the process of solving the equation. If at any point the decimals are rounded, the accuracy will decrease. As for the accuracy of machines, if the values are inserted into the calculator without any rounding throughout the whole calculation, the solution will definitely be close to perfect. As I worked through this assignment, I have been able to gain more knowledge about logarithm and the logic behind the equations. At first, I was sure I could not solve complex equations like the one above, yet after learning how to simplify the equation it made it a lot easier to solve. By breaking down the equation into simpler numbers, the equation becomes a whole new level of difficulty. In conclusion, the process in solving for the log equation manually was a great way to understand the concept of log yet, the invention of the calculators has brought more accuracy to mathematics.

References:

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One thought on “Journal Entry 10: Logarithms before calculators

  1. Yurika you have given thorough consideration to the invention of logs and their place in society. You have discussed how levels of accuracy could be improved, and why logs were such a good invention.
    Your working on the problem is correct, clear, and easy to follow. You have explained the steps in a good level of detail and have used correct notation.
    Nice work to finish the year on a very good note!
    I had intended for you to have an initial calculation, then from there take the log, and at the end find the anti-log, but this is minor in the scheme of how much you have contributed here.

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