BASIC CONCEPTS OF LOGARITHMS


Its been quite a while since i last updated, that was due to some of my busy academic schedules. For the next few weeks i will be giving online tutorials in elementary mathematics for the students of matriculation program of the Usmanu Danfodiyo University, Sokoto.

Today we start our tutorial by revising the laws of indices and then applying these laws in evaluating the problems of Logarithms.

The following laws of indices are true for all non-zero values of $a$, $b$, and $x$.

(1) $x^a\times{x^b}=x^{a+b}$

(2) $\frac{x^a}{x^b}=x^{a-b}$

(3) $x^0=1$

(4) $x^{-a}=\frac{1}{x^a}$

(5) $(x^a)^b=x^{ab}$

(6) $x^{\frac{1}{a}}=\sqrt[a]{x}$

(7) $x^{\frac{a}{b}}=\sqrt[b]{x^a}$
Evaluating Numbers as Powers

We can prove very simply that every positive real number $N$ can be expressed as a power of a given positive real number $b$, if $b$ is not equal to one. This fact makes it possible to simplify computations by the use of Logarithms.

Example
Simplify $\frac{(27).(81).(27^4)}{(243)(9)(243)}$

Solution
To simplify this problem is quite easy, because it is clear that each number is an integral power of $3$. Therefore, we re-write the given expression as
$\frac{3^3.3^4.(3^3)^4}{3^5.3^2.3^5}$
by the first law of indices.
$\frac{3^{3+4+12}}{3^{5+2+5}}=\frac{3^{19}}{3^{12}}=3^7$

WHAT IS A LOGARITHM?
The logarithm of a number say $N$ to a given base $b$ is the power $P$ to which the base must be raised to equal the number. Mathematically,
$P=\log_bN$ is equivalent to $N=b^P$
For example we can write $8=2^3$ as a log expression of the form $3=\log_28$
Logarithms to the base $10$ are called  common or Briggsian. Examples are
$10=10^1\Rightarrow{\log10=1}$
$1000=10^3\Rightarrow\log1000=3$
$0.1=10^{-1}\Rightarrow\log0.1=-1$
$0.001=10^{-3}\Rightarrow\log0.001=-3$ 

PARTS OF A LOGARITHM

To find the common logarithm of a number, we can express it as a scientific notation $N=a(10^x)$ with $0<a<10$ i.e the values of $a$ is between $1$ and $9$. e.g the number $7340$ is $7.34\times{10^3}$.
Logarithm of every number has two parts namely, the characteristics and the Mantissa , the characteristic is the power of $10$ and the Mantissa is the log of $a$.

WHAT ARE THE LAWS OF LOGARITHM?
The following are the basic laws of logarithm.

Product Law
The logarithm of a product of two numbers $A$ and $B$ is equal to the sum of their logarithms. i.e $\log(AB)=\log{A}+\log{B}$.

Example
Evaluate $\log(25)(0.137)=\log{25}+\log{0.137}$ using calculator, this yields\\
$=1.39379+0.1367-1=0.5346$

Quotient Law
The Logarithm of a quotient of two numbers $A$ and $B$ is equal to the difference of their logarithms $\log\frac{A}{B}=\log{A}-\log{B}$

Example
Evaluate $\log\frac{452.9}{0.00668}=\log{452.9}-\log{0.00668}$ using calculator\\
$=4.8312$

Power Rule
The Logarithm of a number $N$ raised to a power $P$ is equal to the product of the power and the logarithm of the number. i.e $\log{N}^p=P\log{N}$.

Example
Evaluate $\log(37.5)^2=2\log{37.5}=2(1.5740)=3.1480$

Example
Evaluate $\log\sqrt[3]{163.2}=\log(163.2)^{\frac{1}{3}}=\frac{1}{3}\log{163.2}=\frac{1}{3}(2.2127)=0.7376$