Mr. Integral
Math notes/Pre-Calculus/Logarithms

Swapping the Base and the Exponent — Cheat sheet

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The three

log⁡axm=mlog⁡ax\log_a x^m = m \log_a x log⁡anx=1nlog⁡ax\log_{a^n} x = \frac{1}{n} \log_a x log⁡anxm=mnlog⁡ax\log_{a^n} x^m = \frac{m}{n} \log_a x

Argument exponent multiplies. Base exponent divides.

Where each comes from

RuleDerivation
log⁡axm=mlog⁡ax\log_a x^m = m \log_a xx=ak⇒xm=akmx = a^k \Rightarrow x^m = a^{km}
log⁡anx=1nlog⁡ax\log_{a^n} x = \dfrac{1}{n} \log_a x(an)y=any=x⇒ny=log⁡ax(a^n)^y = a^{ny} = x \Rightarrow ny = \log_a x

The method

  1. Write the base and the argument as powers of one common number.
  2. Divide the argument exponent by the base exponent.
  3. Check by raising the base back up.
ExpressionAs powersValue
log⁡285\log_2 8^52152^{15} in the argument1515
log⁡394\log_3 9^4383^8 in the argument88
log⁡48\log_4 8log⁡2223\log_{2^2} 2^332\dfrac{3}{2}
log⁡927\log_9 27log⁡3233\log_{3^2} 3^332\dfrac{3}{2}
log⁡84\log_8 4log⁡2322\log_{2^3} 2^223\dfrac{2}{3}
log⁡832\log_8 32log⁡2325\log_{2^3} 2^553\dfrac{5}{3}
log⁡2781\log_{27} 81log⁡3334\log_{3^3} 3^443\dfrac{4}{3}
log⁡168\log_{16} 8log⁡2423\log_{2^4} 2^334\dfrac{3}{4}
log⁡93\log_9 \sqrt{3}log⁡3231/2\log_{3^2} 3^{1/2}14\dfrac{1}{4}

Bases below 1

A fractional base is a negative power, so the value turns negative.

ExpressionAs powersValue
log⁡1/28\log_{1/2} 8log⁡2−123\log_{2^{-1}} 2^3−3-3
log⁡1/39\log_{1/3} 9log⁡3−132\log_{3^{-1}} 3^2−2-2
log⁡1/48\log_{1/4} 8log⁡2−223\log_{2^{-2}} 2^3−32-\dfrac{3}{2}

Traps

  • log⁡axm\log_a x^m is not (log⁡ax)m(\log_a x)^m. Compare log⁡243=6\log_2 4^3 = 6 with (log⁡24)3=8(\log_2 4)^3 = 8.
  • The base exponent goes underneath: log⁡anx=1nlog⁡ax\log_{a^n} x = \dfrac{1}{n} \log_a x, never nlog⁡axn \log_a x.
  • Roots are fractional powers: log⁡4163=4/32=23\log_4 \sqrt[3]{16} = \dfrac{4/3}{2} = \dfrac{2}{3}.
  • The strict form is log⁡ax2=2log⁡a∣x∣\log_a x^2 = 2 \log_a |x|, because xx may be negative.