# Algebra

The various topics on Algebra will be discussed here.

## Logarithms:

Logarithms are used to express exponential equations in another way. Let a>0 (≠1) and M>0 be such that

ax = M.

Here the number x is called the logarithm of M with base a and we write it mathematically as follows:

x = loga M

log28 : Value of log 8 to the base 2 is equal to 3.

log216 : The value of log 16 base 2 is equal to 4.

log416 : Value of log 16 to the base 4 is equal to 2.

log816 : Value of log 16 to the base 8 is equal to 4/3.

log√216: Value of log 16 base √2 equals 8.

log327 : The value of log 27 base 3 is equal to 3.

log√5125: Value of log 125 base √5 equals 6.

## Complex Numbers:

A complex number z can be expressed as z=a+ib where a and b are real numbers and i=√-1 is the imaginary complex number.

Square root of i

Square root of negative i

A quadratic equation is an equation of degree 2 and it is of the form: ax2+bx+c=0 where a, b, and c are real numbers and a is not equal to 0. The value of x is called the root of the given equation. The root can be computed here:

The nature of its roots is determined by the discriminant D=b2-4ac.

## Binomial Theorem:

For two real numbers a and b and for a natural number n, we get the expansion of (a+b)n by using the binomial theorem:

(a+b)n = an + nC1 an-1b + nC2 an-2b2 + … + bn,

where nCk is defined by

nCk = $\dfrac{n!}{k! (n-k)!}$.

## Determinant:

Reciprocal of 0 | 1/0 is equal to