Mr. Integral
Math notes/Pre-Calculus/Logarithms

The Four Basic Rules — Cheat sheet

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The four

log⁡a1=0\log_a 1 = 0 log⁡aa=1\log_a a = 1 log⁡aan=n\log_a a^n = n alog⁡ax=xa^{\log_a x} = x

The third contains the first two: set n=0n = 0 and n=1n = 1.

The method

Write the argument as a power of the base, then read the exponent off.

LogarithmAs a powerValue
log⁡232\log_2 32252^555
log⁡264\log_2 64262^666
log⁡3181\log_3 \dfrac{1}{81}3−43^{-4}−4-4
log⁡22\log_2 \sqrt{2}21/22^{1/2}12\dfrac{1}{2}
log⁡515\log_5 \dfrac{1}{\sqrt{5}}5−1/25^{-1/2}−12-\dfrac{1}{2}
log⁡41\log_4 1404^000
log⁡77\log_7 7717^111

Roots and fractions are not special cases:

an=a1/n,\sqrt[n]{a} = a^{1/n}, 1an=a−n\frac{1}{a^n} = a^{-n}

Cancelling

alog⁡ax=xa^{\log_a x} = x only when the bases match.

3log⁡37=73^{\log_3 7} = 7 but\text{but} 2log⁡37 simplifies to nothing2^{\log_3 7} \text{ simplifies to nothing}

When the argument is not a whole-number power

Go back to index form and put both sides on a common base.

log⁡48=y  ⇒  4y=8  ⇒  22y=23  ⇒  y=32\log_4 8 = y \;\Rightarrow\; 4^y = 8 \;\Rightarrow\; 2^{2y} = 2^3 \;\Rightarrow\; y = \frac{3}{2}

Traps

  • log⁡a1=0\log_a 1 = 0, not 11. This is the most-swapped pair with log⁡aa=1\log_a a = 1.
  • Different bases cannot be combined. Evaluate each term, then add.
  • 3log⁡34+1=3log⁡34⋅3=4⋅3=123^{\log_3 4 + 1} = 3^{\log_3 4} \cdot 3 = 4 \cdot 3 = 12. Split the exponent first.