Many equations can be simplified by expanding logarithms. The term "expanding logarithms" does not refer to logarithms that expand but rather to a process by which one mathematical expression is substituted for another according to specific rules. There are three such rules. Each of them corresponds to a particular property of exponents because taking a logarithm is the functional inverse of exponentiation: log_{3}(9) = 2 because 3^{2}= 9.
The most common rule for expanding logarithms is used to separate products. The logarithm of a product is the sum of the respective logarithms: log_{a}(x*y) = log_{a}(x) + log_{a}(y). This equation is derived from the formula a^{x} * a^{y} = a^{x+y}. It can be extended to multiple factors: log_{a}(x*y*z*w) = log_{a}(x) + log_{a}(y) + log_{a}(z) + log_{a}(w).
Raising a number to a negative power is equivalent to raising its reciprocal to a positive power: 5^{-2} = (1/5)^{2} = 1/25. The equivalent property for logarithms is that log_{a}(1/x) = -log_{a}(x). When this property is combined with the product rule, it provides a law for taking the logarithm of a ratio: log_{a}(x/y) = log_{a}(x) – log_{a}(y).
The final rule for expanding logarithms relates to the logarithm of a number raised to a power. Using the product rule, one finds that log_{a}(x^{2}) = log_{a}(x) + log_{a}(x) = 2*log_{a}(x). Similarly, log_{a}(x^{3}) = log_{a}(x) + log_{a}(x) + log_{a}(x) = 3*log_{a}(x). In general, log_{a}(x^{n}) = n*log_{a}(x), even if n is not a whole number.
These rules can be combined to expand log expressions of more complex character. For example, one can apply the second rule to log_{a}(x^{2}y/z), obtaining the expression log_{a}(x^{2}y) – log_{a}(z). Then the first rule can be applied to the first term, yielding log_{a}(x^{2}) + log_{a}(y) – log_{a}(z). Lastly, applying the third rule leads to the expression 2*log_{a}(x) + log_{a}(y) – log_{a}(z).
Expanding logarithms allows many equations to be solved quickly. For example, someone might open a savings account with $400 US Dollars. If the account pays 2 percent annual interest compounded monthly, the number of months required before the account doubles in value can be found with the equation 400*(1 + 0.02/12)^{m} = 800. Dividing by 400 yields (1 + 0.02/12)^{m} = 2. Taking the base-10 logarithm of both sides generates the equation log_{10}(1 + 0.02/12)^{m} = log_{10}(2).
This equation can be simplified using the power rule to m*log_{10}(1 + 0.02/12) = log_{10}(2). Using a calculator to find the logarithms yields m*(0.00072322) = 0.30102. One finds upon solving for m that it will take 417 months for the account to double in value if no additional money is deposited.