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.