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What Are Expanding Logarithms?

By David Isaac Rudel
Updated May 21, 2024
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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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