A logarithm is the inverse of exponentiation. The statement \(\log_b(x) = y\) means exactly the same thing as \(b^y = x\), answering the question: to what power must the base \(b\) be raised to produce \(x\)? Two specific bases are so common that they have their own notation: the common logarithm uses base \(10\) and is written as \(\log(x)\), so \(\log(1000) = 3\) because \(10^3 = 1000\); the natural logarithm uses base \(e\) (approximately \(2.718\)) and is written as \(\ln(x)\). The natural logarithm appears constantly in calculus and in models of exponential growth and decay. To solve an equation like \(\log_2(x) = 5\), you convert from logarithmic to exponential form, obtaining \(x = 2^5 = 32\).
Logarithms obey three key rules that mirror the rules for exponents. The product rule states that \(\log_b(MN) = \log_b(M) + \log_b(N)\); the quotient rule states that \(\log_b(M / N) = \log_b(M) - \log_b(N)\); and the power rule states that \(\log_b(M^n) = n \cdot \log_b(M)\). These rules allow complicated logarithmic expressions to be expanded or condensed. The change of base formula, \(\log_b(x) = \frac{\log_a(x)}{\log_a(b)}\), makes it possible to evaluate any logarithm on a calculator that only has common or natural log keys, since \(\log_b(x) = \frac{\ln(x)}{\ln(b)}\).
Sequences are ordered lists of numbers that often follow a pattern. An arithmetic sequence is one in which each term differs from the previous term by a constant called the common difference \(d\); the \(n\)th term is given by \(a_n = a_1 + (n - 1)d\). For example, the sequence \(3, 7, 11, 15, \ldots\) has common difference \(4\), so its tenth term is \(a_{10} = 3 + 9 \cdot 4 = 39\). The sum of the first \(n\) terms of an arithmetic series is \(S_n = \frac{n(a_1 + a_n)}{2}\), or equivalently \(S_n = \frac{n}{2}[2a_1 + (n - 1)d]\), which can be remembered as the number of terms times the average of the first and last terms.
A geometric sequence, by contrast, is one in which each term is obtained by multiplying the previous term by a constant called the common ratio \(r\); the \(n\)th term is \(a_n = a_1 \cdot r^{n-1}\). The sum of the first \(n\) terms of a geometric series is \(S_n = \frac{a_1(1 - r^n)}{1 - r}\), provided \(r \neq 1\). When the absolute value of the common ratio is less than one, the series converges as more and more terms are added, and the infinite geometric series has a finite sum given by \(S = \frac{a_1}{1 - r}\). For the series \(8 + 4 + 2 + 1 + \ldots\), the first term is \(8\) and the ratio is \(\frac{1}{2}\), so the sum is \(\frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16\).