A logarithmic expression can be rewritten using logarithms with any positive base other than 1 using the equation .

The number is an irrational number with an infinite decimal expansion that starts , which is used in finance and science as the base for an exponential function.

A sequence in which each term is a constant times the previous term.

The time it takes for a material to decay to half of its original value. For example, the half-life of Carbon-14 is about 5,730 years. This means that if an object started with 10 micrograms of Carbon-14, then after 5,730 years it will have 5 micrograms of Carbon-14 left.

The logarithm to base 10 of a number , written , is the exponent you raise 10 to get , so it is the number that makes the equation true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. is the number that makes the equation true. The logarithm to the base is called the natural logarithm, and is written .

A logarithmic function is a constant multiple of a logarithm to some base, so it is a function given by where is any number and is a positive number (10, 2, or in this course). The graph of a typical logarithmic function is shown. Although the function grows very slowly, the graph does not have a horizontal asymptote.

The natural logarithm of , written , is the log to the base of . So it is the number that makes the equation true.

A logarithm with an argument raised to a power is equivalent to the power multiplied by the logarithm of the argument with a power of 1.

The sum of two logarithms with the same base is equivalent to a logarithm with the same base of the product of the arguments.

The difference of logarithms with the same base is equivalent to a logarithm of the quotient of the arguments.