Logarithms

 

 

What is a logarithm?

 

Question: How do we reduce an exponential growth of grass?

Answer: Using a "ln"-mower!

 

The above joke embodies the idea of logarithms. Logarithms are initially developed to reduce large numbers into smaller numbers, so that operations such as multiplication and division can be performed more efficiently. Do not forget that this is way before the calculator is developed! When translated literally from the Chinese language, logarithm means reduced number.

 

The logarithmic form is another way of expressing the exponential form, this time in much smaller numbers. Like surds, most logarithms are irrational numbers. In mathematical notation,

                                                 

              provided a is positive.

log_ay is read as "logarithm of y to base a".

 

 

Express the following in index form:

     (i)   log₃a = y                  3^y = a

     (ii)  log_ab = a                  a^a = b

     (iii) log₃x = 4                  3⁴ = x

 

Express the following in logarithmic form:

     (i)    4^a = y                     log₄y = a

     (ii)   x^b = 7                     log_x7 = b

     (iii)  k⁵ = x                     log_kx = 5

 

 

Common & Natural Logarithms

 

Logarithms to base 10 are known as common logarithms. They are often abbreviated as lg.

 

e is a naturally occurring number, one of the fundamental numbers in mathematics. It is approximated to 2.71828 (5 decimal places). Logarithms to base e are natural logarithms. It is often abbreviated as ln (hence the joke).

 

 

Practical Applications of Logarithms

 

The idea of logarithms is not just an abstract mathematical concept. It has many practical uses. In the past, before the development of the calculator, mathematicians reduce large numbers to logarithms, and then solve questions by referring to a log table, where  the commonly used logarithms are calculated and tabulated. Today, this method is nearly obsolete. Two modern day uses of logarithms are the pH scale and Richter scale.

 

The pH scale is used in chemistry to determine the acidity or alkalinity of a solution. The scale ranges from 1 to 14, with 1 being the most acidic and 14 the most alkaline. The difference in strength of an acid of pH 1 and that of pH 2 is not twofold, but tenfold. Why?

 

The pH scale is actually a logarithm in the form:

                       log₁₀[concentration of hydrogen ions]

Thus, pH 1 = log₁₀10, and pH 2 = log₁₀100.

 

The Richter scale is used to determine the strength of the ground movement. The larger the number, the more violent the movement. The Richter scale is commonly associated with earthquakes. Similar to the pH scale, an earthquake of magnitude 7 on the scale is ten times stronger than an earthquake of magnitude 6. Again, this is because the Richter scale is actually a logarithm:

                       log₁₀[measurement of movement of the earth]

 

Thus you can see that logarithms are useful in the real world.