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Expanding logarithms involves replacing expressions according to specific rules. The three rules involve separating products, taking the logarithm of a ratio, and taking the logarithm of a number raised to a power. These rules can be combined to solve complex equations, such as finding the time it takes for a savings account to double in value.
Many equations can be simplified by expanding logarithms. The term “expanding logarithms” does not refer to expanding logarithms but rather to a process by which one mathematical expression is replaced by 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 their 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 several 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 last rule for expanding logarithms involves taking the logarithm of a number raised to the power. Using the product rule, we find that loga(x2) = loga(x) + loga(x) = 2*loga(x). Similarly, log(x3) = log(x) + log(x) + log(x) = 3*log(x). In general, loga(xn) = n*loga(x), even if n is not an integer.
These rules can be combined to expand register expressions of a more complex character. For example, the second rule can be applied to loga(x2y/z), resulting in the expression loga(x2y) – loga(z). So the first rule can be applied to the first term, obtaining log(x2) + log(y) – log(z). Finally, the application of the third rule leads to the expression 2*loga(x) + loga(y) – log(z).
Logarithm expansion allows you to solve many equations quickly. For example, someone could open a savings account with US$400. If the account pays 2% annual interest compounded monthly, the number of months it takes before the account doubles in value can be found by 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 gives the equation log10(1 + 0.02/12)m = log10(2).
This equation can be simplified using the power rule in m*log10(1 + 0.02/12) = log10(2). Using a calculator to find logarithms gives m*(0.00072322) = 0.30102. Solving for m, it turns out that it will take 417 months for the account to double in value if no additional money is deposited.
