# Investigation: log=bc

John Napier, the founder of logarithm.

http://www.electricscotland.com/history/other/john_napier.htm

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