Further Pure Mathematics

Core

# Logarithmic functions

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Logarithmic functions

Logarithmic functions

Basic Definition and Properties

- A
**logarithm**is the power to which a certain number, called the base, must be raised to obtain a given number. - The expression is written as
**log_b(a) = n**where b is the base, a is the number and n is the power. - Logarithmic functions are the
**inverses**of exponential functions.

Basic Rules of Logarithms

**Product rule**: The log of a product is the sum of the logs of its factors, i.e.,**log_b(a * c) = log_b(a) + log_b(c)**.**Quotient rule**: The log of a quotient is the difference between the logs of the numerator and the denominator, i.e.,**log_b(a / c) = log_b(a) - log_b(c)**.**Power rule**: The log of an exponent is the exponent times the log of the base, i.e.,**log_b(a^n) = n * log_b(a)**.

Change of Base Formula

- Any logarithm can be computed using any other base through the change of base formula, as long as both the bases are positive and not equal to 1.
- The
**change of base formula**is written as**log_b(a) = log_c(a) / log_c(b)**where a, b, c are positive numbers and b ≠ 1.

Natural Logarithm

- The
**natural logarithm**or**ln**is a logarithm in the base e, where e is an irrational and transcendental number approximately equal to 2.71828. - Natural logarithms have similar properties to those mentioned above for basic logarithms.

Solving Logarithmic Equations

- To solve equations involving logarithms, use the
**rules of logarithms**to simplify the equations. - If the logs in the equation have the same base, you can set the expressions inside the logs equal to each other and solve the resulting equation.
- In some cases, it may be helpful to rewrite the logarithmic equation as an exponential equation.

Relationship with Exponential Functions

- The set of all exponential functions is the inverse of the set of all logarithmic functions, and vice versa.
- The graph of a logarithm function is a reflection of the graph of the corresponding exponential function over the line y = x.