Logarithmic and exponential functions

Logarithmic Functions

Logarithms: An exponent to which a base must be raised to produce a given number. Written as log(base)(number).
Common logarithm: A logarithm with base 10, often written simply as log(number) without specifying the base.
Natural logarithm: A logarithm with base e (approximately equal to 2.71828), written as ln(number).
Logarithmic Properties: Basic logarithm properties include change of base formula, product rule, quotient rule, and power rule.
Inverse of logarithm: The inverse of a logarithmic function is an exponential function, and vice versa.
Change of base rule: Any logarithm can be computed using another base log by using the formula log(base)(number) = log(newbase)(number) / log(newbase)(base).
Solving logarithmic equations: To solve some logarithmic equations, use the property that if a = b then log(a) = log(b).

Exponential function: A function written as f(x) = b^x, where the base b is a positive real number other than 1.
Exponential growth and decay: Exponential functions where the base is larger or smaller than 1, respectively. They occur commonly in real-world applications.
Properties of exponents: Basic exponents rules include product of powers, quotient of powers, power of a power, power of a product, and power of a quotient.
Halflife and doubling time: Examples of exponential decay and growth, respectively. Halflife is the time for a quantity to decrease by half, while doubling time is the time for it to double.
Continuously compounded interest: An example of an exponential growth function where interest is calculated and added to the account’s total continuously.
Solving exponential equations: To solve some exponential equations, use the property that if a = b then ln(a) = ln(b).

Exponential equations: Equations in which the variable x occurs in an exponent.
Logarithmic equations: Equations involving one or more logarithmic expressions.
Solving exponential and logarithmic equations: These equations can be solved by using the laws of exponents or logarithms, or by converting between exponential and logarithmic form.
Graphs of exponential and logarithmic functions: Exponential functions increase/decrease rapidly, whereas logarithmic functions increase/decrease slowly. These properties are reflected in their graphs.
Transformations of exponential and logarithmic functions: Includes translations, reflections, and dilations of their graphs. These transformations follow the same rules as for polynomial functions.