# What Are Expanding Logarithms?

David Isaac Rudel
David Isaac Rudel

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: loga(x*y) = loga(x) + loga(y). This equation is derived from the formula ax * ay = ax+y. It can be extended to multiple factors: loga(x*y*z*w) = loga(x) + loga(y) + loga(z) + loga(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 loga(1/x) = -loga(x). When this property is combined with the product rule, it provides a law for taking the logarithm of a ratio: loga(x/y) = loga(x) – loga(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 loga(x2) = loga(x) + loga(x) = 2*loga(x). Similarly, loga(x3) = loga(x) + loga(x) + loga(x) = 3*loga(x). In general, loga(xn) = n*loga(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 loga(x2y/z), obtaining the expression loga(x2y) – loga(z). Then the first rule can be applied to the first term, yielding loga(x2) + loga(y) – loga(z). Lastly, applying the third rule leads to the expression 2*loga(x) + loga(y) – loga(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.

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