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
4 then we can write
then we can write
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
Example so we can write
Example in logarithm form
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.
̅.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.