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Pure · Exponentials & logarithms

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The laws of logarithms

The product, quotient and power laws — turning sums and differences of logs into a single log (and back) — plus log_a a = 1, log_a 1 = 0 and the reciprocal rule.

Pure · Exponentials & logarithms

The laws of logarithms

The product, quotient and power laws — turning sums and differences of logs into a single log (and back) — plus log_a a = 1, log_a 1 = 0 and the reciprocal rule.

Why it works

The log laws are just the index laws seen through the inverse. Because am×an=am+na^m \times a^n = a^{m+n}, adding logs corresponds to multiplying, and so on. For the same base aa:

log⁡ax+log⁡ay=log⁡a(xy)(product law)\log_a x + \log_a y = \log_a(xy) \quad\text{(product law)} log⁡ax−log⁡ay=log⁡a ⁣(xy)(quotient law)\log_a x - \log_a y = \log_a\!\left(\tfrac{x}{y}\right) \quad\text{(quotient law)} log⁡a(xk)=klog⁡ax(power law)\log_a (x^k) = k\log_a x \quad\text{(power law)}

From the power law with k=−1k = -1 comes the handy reciprocal rule log⁡a ⁣(1x)=−log⁡ax\log_a\!\left(\tfrac1x\right) = -\log_a x, and recall the two values log⁡aa=1\log_a a = 1, log⁡a1=0\log_a 1 = 0.

Two cautions that catch people out:
  • The laws only combine logs of the same base. You cannot merge
log⁡25+log⁡35\log_2 5 + \log_3 5.
  • A log of a sum is not the sum of logs. $\log_a(x + y) \neq \log_a x +
\log_a y.Theproductlawisabout. The product law is about \log_a(xy),not, not \log_a(x+y)—keepthebracketsstraight.Thepowerlawistheworkhorse:itiswhatletsyoubringanunknown∗downfromtheexponent∗,whichishoweveryequationlike — keep the brackets straight. The power law is the workhorse: it is what lets you bring an unknown *down from the exponent*, which is how every equation like 3^x = 20$ gets solved.