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

We write

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.

Log table-