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Before calculators existed, 17th-century astronomers spent grueling weeks multiplying ten-digit numbers by hand until Scottish mathematician John Napier invented logarithms in 1614 to transform tedious multiplication into simple addition. A logarithm answers an exponent question: log⁑a(x)=y\log_a(x) = y simply means that base aa raised to the power yy equals xx (ay=xa^y = x).

The Product Law

The product law of logarithms states that the logarithm of a product equals the sum of the individual logarithms: log⁑a(xy)=log⁑a(x)+log⁑a(y)\log_a(xy) = \log_a(x) + \log_a(y), where a>0a > 0, aβ‰ 1a \neq 1, and x,y>0x, y > 0.
For the formal proof, let u=log⁑a(x)u = \log_a(x) and v=log⁑a(y)v = \log_a(y), which implies x=aux = a^u and y=avy = a^v. Multiplying these gives xy=auβ‹…av=au+vxy = a^u \cdot a^v = a^{u+v}, and converting back to logarithmic form yields log⁑a(xy)=u+v=log⁑a(x)+log⁑a(y)\log_a(xy) = u + v = \log_a(x) + \log_a(y).
For the formal proof, let u=log⁑a(x)u = \log_a(x) and v=log⁑a(y)v = \log_a(y), meaning x=aux = a^u and y=avy = a^v. Multiplying these gives xy=auβ‹…av=au+vxy = a^u \cdot a^v = a^{u+v}, and converting back to logarithmic form yields log⁑a(xy)=u+v=log⁑a(x)+log⁑a(y)\log_a(xy) = u + v = \log_a(x) + \log_a(y).

Expanding and Combining

Through bidirectional application, the law allows expanding composite products into separate terms (log⁑a(xy)β†’log⁑a(x)+log⁑a(y)\log_a(xy) \to \log_a(x) + \log_a(y)) and condensing separate terms into a single logarithm (log⁑a(x)+log⁑a(y)β†’log⁑a(xy)\log_a(x) + \log_a(y) \to \log_a(xy)).
Additionally, the base consistency requirement strictly demands that all logarithmic terms share the exact same base aa. Expressions with different bases, such as log⁑2(x)+log⁑3(y)\log_2(x) + \log_3(y), cannot be combined directly using the product law.
A common mistake is writing log⁑a(x+y)=log⁑a(x)+log⁑a(y)\log_a(x + y) = \log_a(x) + \log_a(y) or multiplying logs together like log⁑a(x)β‹…log⁑a(y)=log⁑a(xy)\log_a(x) \cdot \log_a(y) = \log_a(xy). Remember: the addition must happen between separate logs of the same base, while the multiplication happens strictly inside a single log's argument.
Furthermore, the base consistency requirement dictates that all logarithmic terms must share the exact same base aa. Expressions with different bases, such as log⁑2(x)+log⁑3(y)\log_2(x) + \log_3(y), cannot be combined directly using the product law.

Key takeaways