CAT 2025 Lesson : Logarithm - Basics of Logarithm

1. Introduction

For most of you who have studied logarithm in school, log and anti-log tables would be the next thing you'd remember. In the absence of a calculator, we applied logarithmic properties and the log/anti-log tables to solve problems with calculations involving huge numbers. These typically involved exponents and multiplication.

If two big numbers need to be multiplied or divided, after taking logarithm of these numbers they need to be added or subtracted. We get the answer by applying anti-log to the result. Students are not tested with arithmetic calculations involving such big numbers using log/anti-log conversions or the associated tables in CAT and other management entrance exams. The questions are on the fundamentals and properties, which are covered here.

Logarithm was very handy a few decades back when computers were not widely available to work with large numbers and functions. The use or the importance of logarithm, however, has not diminished with technological advancement. It is still a key part of our lives and we keep hearing these numbers, but probably do not associate it with logarithm. Some of these are listed below.

Richter Scale: Earthquake measuring $7.7$ on the richter scale struck Northern India in October $2015$ and another measuring $6.7$ struck Northeast India in January $2016$. Richter scale uses a base$-10$ logarithmic scale which describes the magnitude of the quake. So, the magnitude of the quake measuring $7.7$ was $10$ times that of the one measuring $6.7$. A logarithmic scale here makes it easier to compare and understand the impact.

Exponential decay and continuous compounding: If we observe that $18.45 \%$ of a certain radioactive substance has decayed over $300$ years, how long will it take for half the substance to decay? If I have increased my wealth by $28 \%$ over the last $200$ days, assuming I continue to grow at the same rate, how long will it take for my wealth to double? Using logarithm is the simplest way to answer both these questions.

Decibels: The power of the sound of an electric razor at $90$ decibels is $10$ times greater than a vacuum cleaner at $80$ decibels or $100,000$ times greater than a refrigerator at $40$ decibels.

1.1 Taking Logarithm of a number

Logarithm of a number with a certain base is the exponent, which the base has to be raised to produce the number. Please find the log operation below.



(Reduce Size)
$16 = 2^{4}$ can be written as $\mathrm{log} _{2} {16} = 4$
$5 = 125^\frac{1}{3}$ can be written as $\mathrm{log} _{125} {5} = \dfrac{1}{3}$
$0.01 = 10^{-2}$ can be written as log$_{10}$ $0.01$ $= -2$

Example 1

If $\mathrm{log} _{3} {x} = 4$ and $\mathrm{log} _{5} {0.04} = y$, then what is $x + y?$

Solution

$\mathrm{log} _{3} {x} = 4$
⇒ $x = 3^{4}$
⇒ $x = 81$

$\mathrm{log} _{5} {0.04} = y$
⇒ $0.04 = 5^{y}$
⇒ $\dfrac{1}{25} = 5^{y}$
⇒ $5^{-2} = 5^{y}$
⇒ $y = -2$

$x + y = 81 - 2 = 79$

Answer: $79$

1.2 Graph of the Logarithmic Function

Note: For any logarithmic function, $\mathrm{log} _{a} {1} = 0$, where $a$ is any defined base. Therefore, all log curves will cut the $x$-axis at $x = 1$.

1.3 Types of Bases

The base in a logarithm can be any number. However, as logarithm is applied for specific purposes, there are three commonly used bases.

1) Common Logarithm: Logarithms with base $10$. These are the most common and have applications in a lot of scientific fields. Decibels and Richter scale use common logarithms. As this is very common, if a logarithm is provided without the base, it is assumed to be common logarithm. If a question has '$\mathrm{log} 2$' mentioned, take this to be $\mathrm{log} _{10} {2}$ and solve the problem.

2) Natural Logarithm: Logarithms with base e. Written as '$\mathrm{log} _{e} {x}$' and also as 'ln $x$'. Has widespread application in science and finance/economics. '$e$' is used in calculations involving continuous compounding.

3) Binary Logarithm: Logarithms with base $\bm{2}$. This has gained prominence in the recent past and has applications in computer science and modern technology.