Logarithms

Since 2⁴ = 16, log₂ 16 = 4.

The logarithm is the inverse of the exponential function. It shows how many times a number must be multiplied by itself to get another number.

Here, a is the base (positive number, cannot be 1), b is the number to be logged (positive), and c is the result, i.e., the exponent.

Important Identities

For all a > 0, if a ≠ 1:

These identities help simplify logarithms and solve equations.

Special Logarithms

The base-10 logarithm is called the common logarithm, denoted by: log. The natural logarithm has base e (Euler's number ≈ 2.718), denoted by: ln.

Logarithmic Equations

Example of a simple logarithmic equation:

This means x is the number such that 2³ = x, so x = 8.

Relationship with Exponential Functions

The logarithm and exponential function are inverses of each other. This means the logarithm 'reverses' the exponentiation.

Practical Applications

Finance: the relationship between exponential and logarithm appears in compound interest calculations.
Sciences: the pH scale in chemistry is based on logarithms.
Computer Science: it appears in measuring algorithm complexity (e.g., O(log n)).

Practice Exercise

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