Calculating Logarithms

A logarithm is a mathematical function closely related to exponentials. In fact, the logarithm is the inverse of the exponential function. The general form is logb(x), which reads "log base b of x." Frequently, log with no base implies base 10 logs log10, and ln refers to the "natural log," loge, where e is an important transcendental number, e = 2.718282 .... In general, to calculate logb(x), you would use a calculator, but knowing the properties of logarithms can help solve particular problems.

The definition of a logarithmic base is logb(b) = 1. The definition of the logarithmic function is if y = b^x, then logb(y) = x. Some other important properties are logb(xy) = logb(x) + logb(y), logb(x/y) = logb(x) – logb(y), and logb(x^y) = ylogb(x). You can use these properties to help you calculate logarithms in different situations.

Sometimes you can quickly calculate logb(x) if you can answer the problem b^y = x. Log10(1,000) = 3 because 10^3 = 1,000. Log4(16) = 2 because 4^2 = 16. Log25(5) = 0.5 because 25^(1/2) = 5. Log16(1/2) = -1/4 because 16^(-1/4) = 1/2, or (1/2)^4 = 1/16. Using logb(xy) formula, log2(72) = log2(8 * 9) = log2(8) + log2(9) = 3 + log2(9). If we estimate log2(9) ~ log2(8) = 3, then log2(72) ~ 6. The actual value is 6.2.

Suppose you know logb(x), but you want to know loga(x). This is called changing bases. Because a^(loga(x)) = x, you can write logb(x) = logb[a^(loga(x))]. Using logb(x^y) = ylogb(x), you can turn this into logb(x) = loga(x)logb(a). By dividing both sides by logb(a), you can solve for loga(x): loga(x) = logb(x) / logb(a). If you have a calculator that does base 10 logs, but you want to know log16(7.3), you can find it by log16(7.3) = log10(7.3)/log10(16) = 0.717.