What Are Expanding Logarithms?

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: log₃(9) = 2 because 3²= 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(x²) = log_a(x) + log_a(x) = 2*log_a(x). Similarly, log_a(x³) = log_a(x) + log_a(x) + log_a(x) = 3*log_a(x). In general, log_a(xⁿ) = 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(x²y/z), obtaining the expression log_a(x²y) – log_a(z). Then the first rule can be applied to the first term, yielding log_a(x²) + 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 log₁₀(1 + 0.02/12)^m = log₁₀(2).

This equation can be simplified using the power rule to m*log₁₀(1 + 0.02/12) = log₁₀(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.