Logarithms - Introduction to logarithms || Importance of Logarithm || Scientific Notation || Laws of Logarithm || Examples of Logarithms

Introduction of logarithms

• Logarithms were introduced by the great Muslim mathematician Abu Muhammad Musa al-Khwarizmi.
• Abu Muhammad Musa al-Khwarizmi first gave an idea of logarithms.

Later on

• Logarithms developed In $17^{th}$ century by John Napier the concept of the logarithm further and developed/prepared tables for logarithms
• These tables utilized the base "$e$" The estimated value of the irrational number $e$ is $2.71828...$.
• The renowned mathematician "Euler" discovered the characteristics of the number "$e$". Therefore, this number is represented by the initial letter of his name.
• John Napier's writings caught the curiosity of Professor Henry Briggs.
• He created base-10 logarithm tables in the amount of $1631.
• Joost Burgi created the antilogarithm table in the year 1620 AD.

Importance of Logarithm

• Using logarithms makes complex and challenging computations easier.
  
  

Scientific Notation

• Scientific notation is a short way to write very large and very small numbers.
• Generally, approximate values of the given numbers are used to write them in scientific notation.
• In the problems of mathematics and other branches of science, we deal with very small or very large numbers.

For examples:

$1.$   $0.000000847$
$2.$   $56,784,234,079$

$3.$   The weight of the electron is

$0.000,000,000,000,000,000,000$, $000,000,000,000,910,905$ $kg$

$4.$   The weight of earth $6,000,000,000,000,000,000,000,000$ $kg$

 

• For the sake of convenience, we write such numbers, by using a special notation which is known as scientific notation.
• In this notation, the given number is expressed as the product of two factors.
• One of which is a number equal to or greater than $1$ but less than $10$, and the other is a power of $10$.
• If the given number is $n$: then in scientific notation $n=s\times 10^m$ where $\le s\lt 10$ and $m$ is an integer.

 

Example:
write the numbers in scientific notation

$i)$   $5373.458$
solution:

• The first factor is obtained by placing a decimal point after the first-hand non-zero digit.
  $5.373458\times 1000$           

• The second factor is a power of $10$ and the exponent is obtained by counting the number of digits that must be passed over to move from the original position of the decimal point to its new position.
  $5.373458\times 10^3$

• If the decimal point moves from left to right then the exponent is negative.
  $0.0004302$

• If the decimal point moves from right to left then the exponent is negative.
  $4300$

       

$ii)$    $89000000$

solution:
$8.9\times 1000000$
$8.9\times 10^7$                    Ans.

$iii)$    $0.0007689$

solution:
$\frac{7.689}{10000}$

$\frac {7.689}{10^4}=7.689\times 10^{-4}$            Ans.

$iv)$    $0.00000015$

solution:
$\frac {1.5}{1000000}=\frac {1.5}{10^7}=1.5\times 10^{-7}$        Ans.

Definition of Logarithm

• Let $a,x,y$ be real numbers such that $a\gt 0$ and $a\ne 0$, if $a^y=x$, then logarithm of $x$ to the base $a$ is $y$.
• It is written as $\log_{a} x=y$.

     $\log_{a} N=x\Longleftrightarrow$ $N=a^x$  
Logarithm form       Exponential form

Example:
$81=3^4\Longleftrightarrow \log_{3} 81=4$

$(36)^{3/2}=216\Leftrightarrow \log_{36} 216=\frac {3}{2}$

$2^{-7}=\frac {1}{128}\Leftrightarrow \log_{2} \frac {1}{128}=-7$

$10^{-2}=0.01\Leftrightarrow \log_{10} 0.01=-2$

Write in exponential form

$\log_{5}25=2\Leftrightarrow 5^2=25$

$\log_{10}0.001=-3\Leftrightarrow 10^{-2}=0.001$

$\log_{2}\frac{1}{8}=-3\Leftrightarrow 2^{-3}=\frac {1}{8}$

• The logarithm of a number is unique
  In other words, a number cannot have two distinct $log$ to the same base.
• If $a=1$ , then the equation $x=a^y$ has $n$ unique solutions
  for example $1^1=1$, $1^2=1$, $1^3=1$, $1^4=1$, etc $y$ may assume any of the values $1,2,3,4,...$
  
• The condition $a\gt 0$ ensures that $x$ is always real.
• If $x=1$, then $1=a^0$ for all values of $a$. Therefore $\log_{a} 1=0$. Whatever the case may be.
  The logarithm of $1$ to any base is zero
• If $y=1$, then $x=a^1\Rightarrow a=a^1$ Therefore $log_{a} a=1$
  The logarithm of the base to itself is $1$
  
  

$\log_{100}{100}=1$,   $\log_{8}{8}=1$,   $\log_{2}{2}=1$,   $\log_{5}{5}=1$

Problem 1: Find the value of $x$ in the following.
$i)$  $\log_{32}{x}=-\frac {1}{5}$
Solution:

$32^{-\frac{1}{5}}=x$

$\frac{1}{(2^5)^{1/5}}=x$

$x=\frac{1}{2}$    Ans

$ii)$  $log_{10} x=-4$

Solution:
$10^{-4}=x$

$x=\frac{1}{10^4}$

$x=\frac{1}{10000}$    Ans.

Problem: 2  Find the value of $a$ in the following

$i)$ $log_{a} 3=\frac{1}{2}$

Solution:
$a^{1/2}=3$

Squaring on both sides

$a=3$  Ans.

$ii)$ $log_{a}{\frac{1}{25}}=-\frac{2}{3}$

Solution:
$log_{a}{\frac{1}{25}}=-\frac{2}{3}$ $\rightarrow a^{-\frac{2}{3}}=\frac{1}{25}$

$a^{-\frac{2}{3}}=25^{-1}$

$a^{\frac{2}{3}}=25$

$a^\frac{1}{3}=5$

$a=125$    Ans.

Problem: 3  Find the value of $y$ in the following

$(i)$ $log_{5}{\frac{1}{5}}=y$

Solution:

$5^y=\frac{1}{5}$

$5^y=5^{-1}$

∴ $log_{a}{\frac{1}{a}}=-1$

$y=-1$    Ans.

$(ii)$ $log_{55} 55=y$

Solution:

$55^y=55$

$y=1$    Ans.

Problem: 4  Find the logarithms

$i)$ $1728$ to base $2\sqrt {3}$

Solution:

Let $log_{2\sqrt {3}} 1728=x$

$(2\sqrt 3)^x=1728$

$(2.3^{1/2})^x=2^6.3^3$

$(2.3^{1/2})^x=2^6.3^{2/2.3}$

$(2.3^{1/2})^x=(2.3^{1/2})^6$

$x=6$    Ans.

$ii)$  $125$ to base $5\sqrt 5$

Solution:

let $log_{5} {\sqrt{5}}125=x$

$(5\sqrt{5})^x=125$

$(5.5^{1/2})^x=5^3$

$(5^{1+1/2})^x=5^3$

$(5\sqrt{5})^x=125$

$(5.5^{1/2})^x=5^3$

$(5^{1+1/2})^x=5^3$

$(5^{3/2})^x=5^{3/2\times 2}$

$(5^3/2)^x=(5^{3/2})^2$

$x=2$    Ans.

Laws of Logarithm

We shall state and prove three laws of logarithms that enable us to simplify numerical calculations.

1. First laws:

For real numbers $m$, $n$, and $a$ $\gt 0$, $\neq 1$.

$log_{a} mn=log_{a} m+log_{a} n$

Proof:

Let $log_{a} m=x$ and $log_{a} n=y$

Exponential Form
$a^x=m$ and $a^y=n$
Multiplying both

Law of indices

$a^x.a^y=mn$

$a^{a+b}=mn$

Logarithm form

$log_{a} {mn}=log_{a} m+log_{a} n$   proved.

The logarithm of the product of two or more two numbers is the sum of their logarithms.

$log_{a} (xyz)=log_{a} x+log_{a} y+log_{a} z$

2. Second Law:

For real numbers $m$, $n$, $a$ and $a\gt 0$, $a\neq 1$

$log_{a}{\frac{m}{n}}=log_{a}{m}-log_{a}{n}$

Proof:

Let $log_{a}{m}=x$ and $log_{a}{n}=y$

$a^x=m$ and $a^y=n$

Dividing $1^{st}$ by $2^{nd}$

Exponential form
$\frac{a^x}{a^y}=\frac{m}{n}$

Index Law
$a^{x-y}=\frac{m}{n}$

Logarithmic form
$log_{a}\frac{m}{n}=x-y$

$log_{a}{\frac{m}{n}}=log_{a} m-log_{a} n$

3. Third Law

For real numbers $m$, $n$, $a$ and $a\gt 0$, $a\neq 1$

$log_{a} m^n=n log_{a} m$
Proof:

Let $log_{a} m=x$

$a^x=m$

$(a^x)^n=m$

$\left( a^x \right)^n=m^n$

$a^{nx}=m^n$

$(a)^{nx}=\left( m^n \right)$

$log_{a}m^n=nx$

$log_{a}m^n=n\text{ }log_{a}m$   proved.

• The logarithm of a number raised to a power $n$ is the product of the exponent $n$ and the logarithm.
  
  

Change of Base in Logarithms

1. The logarithm's change of base rule is as follows:

$log_{a}n=\frac{log_{b}\text{ }n}{log_{b}\text{ }a}$

Proof:

Let $log_{a}{n}=x$

$a^x=n$

Taking $log$ on both sides to the base $b$

we have

$log_{b}{a^x}=log_{b}{n}$

$x log_{b}{a}=log_{b}{n}$

$x=\frac{log_{b}n}{log_{b}a}$    Proved.

$2.$  Special case:

$log_{a}{b}.log_{b}{a}=1$

Proof:

let $log_{a}{b}=x$

$a^x=b$

Taking $log$ on both sides to the base $b$

$log_{b}{a^x}=log_{b}{b}$

[Since $log_{b}{b}=1$]

$x\text{ } log_{b}{a}=1$

$log_{a}{b}.log_{b}{a}=1$

3. Third Law

For real numbers $m$, $n$, $a$ and $a\gt 0$, $a\neq 1$

$log_{a}{m^n}=nlog_{a}{m}$

Proof:

let $log_{a}{m}=x$

$a^x=m$

$\left( a^x \right)^n=m^n$