Three Key Properties

Algebra 2 · Axiom Academy

LESSON Three Key Logarithm Properties Product, quotient, and power — each rule is an exponent law in disguise, because a logarithm just recovers an exponent. The Foundation: A Log Is an Exponent Logarithms and exponents are inverse operations. Writing an exponential equation and its logarithmic form side by side shows they carry exactly the same information: Read left to right you raise b to an exponent; read right to left the logarithm recovers that exponent. Every rule below is this one idea applied to a specific exponent law. Why does this work? It comes straight from the exponent rule : multiplying powers adds their exponents, so multiplying inside a log adds the logs. Let and , so b^ x =M and b^ y =N . Why does this work? It comes from the exponent rule : dividing powers subtracts their exponents, so dividing inside a log subtracts the logs. Why does this work? It comes from the exponent rule : a power of a power multiplies the exponents. Raising M to the p repeats the multiplication p times, so its exponent is added p times — that is, multiplied by p . Each logarithm rule mirrors one exponent law — the logarithm simply turns the operation down one level: You've seen that the Product, Quotient, and Power rules are the exponent laws in disguise — multiply becomes add, divide becomes subtract, and a power becomes a product. Scroll up to revisit any step.

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