MATHEMATICS

Logarithm Rules

The base b logarithm of a number is the exponent that we need to raise the base in order to get the number.

When b is raised to the power of y is equal x:

b^y = x

Then the base b logarithm of x is equal to y:

log_b(x) = y

For example when:

2⁴ = 16

Then

log₂(16) = 4

The logarithmic function,

y = log_b(x)

is the inverse function of the exponential function,

x = b^y

So if we calculate the exponential function of the logarithm of x (x>0),

f (f ^-1(x)) = b^logb^(x) = x

Or if we calculate the logarithm of the exponential function of x,

f ^-1(f (x)) = log_b(b^x) = x

Natural logarithm is a logarithm to the base e:

ln(x) = log_e(x)

When e constant is the number:
$e=\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^x = 2.718281828459...$
or

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:

x = log^-1(y) = b^y

The logarithmic function has the basic form of:

f (x) = log_b(x)

$e=\lim_{x\rightarrow 0 }\left ( 1+ \right x)^\frac{1}{x}$