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 \(x\), written \(\log_{10}(x)\), is the exponent you raise 10 to get \(x\), so it is the number \(y\) that makes the equation \(10^y = x\) true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. \(\log_2(x)\) is the number \(y\) that makes the equation \(2^y = x\) true. The logarithm to the base \(e\) is called the natural logarithm, and is written \(\ln(x)\).

A logarithmic function is a constant multiple of a logarithm to some base, so it is a function given by \(f(x) = k \log_{a}(x)\) where \(k\) is any number and \(a\) is a positive number (10, 2, or \(e\) 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 \(x\), written \(\ln(x)\), is the log to the base \(e\) of \(x\). So it is the number \(y\) that makes the equation \(e^y = x\) 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.

\(\log_a \left( b^c \right) = c \boldcdot \log_a b\)

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

\(\log_a (b) + \log_a (c) = \log_a (b \boldcdot c)\)

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

\(\log_a (b) - \log_a (c) = \log_a \left( \frac{b}{c} \right)\)