Investigation: log=bc

John Napier, the founder of logarithm.

Even before calculators existed, people used logarithms to find out solutions to real life problems. Log, in simple terms, is the inverse of exponents, and are mainly used to express numbers with large domains. According to A Review of Logarithms (n.d), logarithm was first invented by John Napier, a Scotsman and Joost Burgi, a Swiss. The logarithm they invented was different from each other and different from the logarithm we use today. Napier published his logarithm in 1614, and Burgi’s logarithm was published in 1620. They both had objective in simplifying mathematical calculations but took different approaches. Napier’s took an algebraic approach, whereas Burgi’s approach was geometric. Neither mathematician had a concept of logarithmic base. Napier defined his logarithm in ratio. The use of exponents in logarithmic equations was first noticed in 1685 by John Wallis. The base to the common logarithm used today was invented through the combined effort of Napier and Henry Biggs in 1624. Today, logarithm is useful in many fields from finance to astronomy. The pH table is a great example of where logarithm is used. The pH table measures the concentration of acid in a material. The table is from 0 to 14, and this seems like a short range. But in reality, 0 is quite different from1, even though only one step is taken. This is similar to a measurement of earthquakes. An earthquake of magnitude 5 is quite different from 6. In chemistry, a solution’s pH is defined by the equation , where t is the hydronium ion concentration in moles per liter. As you can see, log is used in this equation. Another example is the measuring of sounds. Sound is measured in a unit called decibels and has a similar relation with logarithm as the pH table. Since each decibel differ so much, log is used in the equation to figure out the decibel of a sound. The equation looks like this: . The value I represent intensity, which is assigned at the beginning. As you can see, by using log, it makes measuring units or a set of number, which has a large domain easier. Before logarithm was invented, people simply had to calculate all mathematics equations using paper and hand. Exponents such as  had to be done on paper as well. Square roots had to be done in your mind, with a best estimate, or using special formulas that took forever to complete. As you can see, in the past where logarithm did not exist, mathematicians would have had to spend a large amount of time on equations that we can solve in minutes today. This explains how much logarithm has helped us today in solving mathematic equations, and its significance in our lives.

The three equations below take you through step by step in how to solve an exponential or square root equation without using a calculator.

Equation 1

Equation 2

Equation 3

Looking at the results I have, using log gives a pretty accurate answer. Since log gives an answer in whole number without any decimals, this will obviously make the answer not accurate to the furthest extent. But still, if the answer given by a calculator is rounded up, it always becomes the answer given by using log. For example, in my first equation, the answer given by the calculator was 338.668. The answer given by using log was 339, and both the answers were very close to each other. One reason why using the log method was not fully accurate was because the decimal was cut off at a point. Just like , people often use 3.14 when calculating, but 3.14 is not accurate because it goes on forever. This will mean that in the past where calculators did not exist, logarithm was not accurate to the full extent. But such minor error would not have made a large impact on the result, so I say that there was not much problem involving accuracy in the past. For these reasons above, my answers were not accurate to the furthest extent, but they were logical and made enough sense to be thought as the actual answer. Logarithm has played a significant role in our lives by making mathematical calculation easier. What struck me was that the calculator answer and manual answer did not have a significant difference. My prediction was that the manually calculated answer would be fairly off compared to the calculator answer, but this assumption was false. Throughout this investigation, I was able to find out how much current technology supports our workings in math, and the effort of past mathematicians who allowed us to have such knowledge and technology.


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