Logarithmic Functions

Business Calculus · Axiom Academy

The inverse of exponential functions—essential for solving growth and decay problems Exponential functions let us calculate future values: "If I invest 1,000 at 5% for 10 years, what do I get?" But what if you want to go backward: "How long until my investment doubles?" This is where logarithms come in—they're the inverse of exponential functions. The logarithm base b of a number x answers the question: "What power do I raise b to in order to get x ?" _ 10 (1000) = 3 because 10^3 = 1000 Logarithms and exponentials "undo" each other, just like addition and subtraction or multiplication and division. If b^y = x , then _b(x) = y . While logarithms can have any positive base, two are especially important in business and mathematics: The natural logarithm (x) uses base e (Euler's number). It's the inverse of e^x : In business calculus, (x) appears constantly because: Continuous compounding uses e^ rt The derivative of (x) is simply x Many growth models use e as the natural base Since logarithms are inverses of exponentials, their graphs are mirror images across the line y = x . Approaches but never touches y-axis You cannot take the logarithm of zero or a negative number! (0) and (-5) are undefined. In business contexts, this makes sense—you can't have negative time or negative quantities in most models. These properties are essential for simplifying expressions and solving equations: Used inverse property on first term, power rule on second term.

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