Laws of Logarithms

Notes:

Terms

The Product Rule:

*loga(x * y) = loga(x) + loga(y)*

  • It’s just a rewritten form of:
    *ar * as = ar + s*

The Quotient Rule:

loga(x / y) = loga(x) - loga(y)

  • It’s just a rewritten form of:
    ar / as = ar - s

The Power Rule:

*loga(xc) = C * loga(x)*

  • It’s just a rewritten form of:
    *(ar)s = ar * s*

Theorems/Formulas

The Change of Base Formula:
logab = ln(b) / ln(a) = log(b) / log(a)

  • Proof:
    • log3(157) = ln(157) / ln(3) ≈ 4.6

AND

  • More Proof:
    • log416 = ln(16) / ln(4) ≈ 2

Logs of reciprocals are opposite:
For any value of x, logax = -loga(1 / x)

  • Proof of Inverse:
    => loga(1 / x) = loga1 - loga(x)

=> loga(1 / x) = 0 - loga(x)

=> -loga(x)

logab = 1 / logba

  • More Proof:
    => log255 = 0.5

=> log525 = 2