Logarithms: Definition Properties

A logarithm is a mathematical tool denoting the exponent a base requires to yield a specific number. Noted as “log_b(x),” where “b” is the base and “x” the value, logarithms simplify calculations by transforming multiplication and exponentiation into addition, playing a pivotal role in various fields for their efficiency in handling complex relationships.

It is a passive real number other than $a^m = n$
We write $m=log_a x$

$10^3 =1000$ 4 then we can write $log_{10} 1000 = 3$
$2^{-3} =\frac{1}{8}$ then we can write $log_2 \frac{1}{8} = -3$

Properties of logarithms

Multiplicative Property: log_b(xy) = log_b(x) + log_b(y)
Divisional Property: log_b(x/y) = log_b(x) – log_b(y)
Power Rule: log_b(x^n) = n * log_b(x)
Change of Base Formula: log_b(x) = log_a(x) / log_a(b)
Logarithm of 1: log_b(1) = 0
Logarithm of Base: log_b(b) = 1
Negative Number Logarithm: log_b(x) is undefined for x ≤ 0
Logarithm of Infinity: log_b(∞) = ∞
Logarithm of Fraction: log_b(1/x) = -log_b(x)
Logarithmic Identity: b^(log_b(x)) = x

$log_a (xy) = log_a x +log_a y$
$log_a (\frac{x}{y}) = log_a x - log_a y$
$log_x x = 1$
Example $2^1=2$ so we can write $log_2 2 = 1$
$log_a 1 = 0$
Example $5^0=1$ in logarithm form $log_5 1 = 0$

$log_a x^p = p(log_a x)$
$log_a x = 1/log_x^a$
$log_a x = \frac{log_b^x}{log_b^a} = \frac{log_⁡x}{log_⁡a}$

Logarithms to the base 10 are known as common logarithms. When base is not mentioned it is taken as 10.
Characteristic – when the number is greater than 1 – the characteristic is one less from the number of digit in the left of the decimal point in the given number.

$(\overline{48}) ̅.48= 1$ $(\overline{6185})$ ̅.41 = 3
When the number is less than 1 one more than the number of zero between the decimal point & the 1st significant digit of the number & it is negative.
0.518 = -1 0.0347 = etc.

Log table-
$10^{.001} = 10^{1/1000} = 1.002305.238$
$10^{0.02} = 1.584293192$
$10^{0.01} = 10^{1/100} = 1.023292992$
$10^{0.6} =3.981071706$
$10^{0.5} = 10^{\frac{1}{2}}= 3.16227766$
$10^{0.3} = 1.995262215$
$10^{0.1} = 10^{\frac{1}{10} = 1.258925412$
$10^{0.7} = 5.0118872336$