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Logarithm Rules


Core Logarithmic Laws

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Product Rule
log⁡a(bc)=log⁡ab+log⁡ac\log_a(bc) = \log_a b + \log_a c
The logarithm of a product is the sum of the logs
Quotient Rule
log⁡a(bc)=log⁡ab−log⁡ac\log_a\left(\frac{b}{c}\right) = \log_a b - \log_a c
The logarithm of a quotient is the difference of the logs
Power Rule
log⁡a(bc)=c⋅log⁡ab\log_a(b^c) = c \cdot \log_a b
The log of a power brings the exponent in front
Root Rule
log⁡a(bn)=1nlog⁡ab\log_a(\sqrt[n]{b}) = \frac{1}{n} \log_a b
A root is just a fractional exponent, so logs turn it into a multiplier
Change of Base Formula
log⁡ab=log⁡cblog⁡ca\log_a b = \frac{\log_c b}{\log_c a}
Converts a log from one base to another

Identity and Constant Rules

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Log of 1
log⁡a1=0\log_a 1 = 0
The log of 1 is always zero in any base
Log of the Base
log⁡aa=1\log_a a = 1
The log of a number to its own base is 1

Inverse and Exponential Relationships

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Inverse Rule
alog⁡ab=ba^{\log_a b} = b and log⁡a(ax)=x\log_a(a^x) = x
Logarithms and exponentials undo each other
Natural Log Identity
ln⁡(ex)=x\ln(e^x) = x and eln⁡x=xe^{\ln x} = x
The natural log and exponential base ee are inverse functions