Mr. Integral
Math notes/Pre-Calculus/Logarithms

The Product Rule — Cheat sheet

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The rule

log⁡ax+log⁡ay=log⁡a(xy)\log_a x + \log_a y = \log_a(xy)

Same base only. Adding logarithms multiplies their arguments.

Why

A logarithm is an exponent, and exponents add when powers multiply.

am⋅an=am+na^m \cdot a^n = a^{m+n}

Worth knowing by sight

SumSingle logarithmValue
log⁡24+log⁡28\log_2 4 + \log_2 8log⁡232\log_2 3255
log⁡25+log⁡4\log 25 + \log 4log⁡100\log 10022
log⁡36+log⁡34.5\log_3 6 + \log_3 4.5log⁡327\log_3 2733
log⁡64+log⁡69\log_6 4 + \log_6 9log⁡636\log_6 3622
log⁡212+log⁡243\log_2 12 + \log_2 \dfrac{4}{3}log⁡216\log_2 1644

Neither term has to be a whole number. Only the product does.

Splitting, the other direction

Break the argument into factors so one of them is a power of the base.

log⁡240=log⁡2(8⋅5)=3+log⁡25\log_2 40 = \log_2(8 \cdot 5) = 3 + \log_2 5

The bases must match

ExpressionCombines?
log⁡53+log⁡57\log_5 3 + \log_5 7yes, log⁡521\log_5 21
log⁡4+log⁡25\log 4 + \log 25yes, both base 1010
ln⁡2+ln⁡6\ln 2 + \ln 6yes, both base ee
log⁡53+log⁡27\log_5 3 + \log_2 7no

Traps

log⁡ax+log⁡ay≠log⁡a(x+y)\log_a x + \log_a y \ne \log_a(x + y) log⁡ax⋅log⁡ay≠log⁡a(xy)\log_a x \cdot \log_a y \ne \log_a(xy)

Given-value questions

Build the argument from the numbers you were given, then convert the product into a sum.

log⁡a2=0.3,  log⁡a5=0.7  ⇒  log⁡a10=0.3+0.7=1\log_a 2 = 0.3,\; \log_a 5 = 0.7 \;\Rightarrow\; \log_a 10 = 0.3 + 0.7 = 1