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Logarithm Rules



Definition and Basic Rules
Fundamental Properties
Base Conversion Rules
Natural Logarithm Properties
Common Logarithm Properties
Advanced Logarithmic Rules
Domain and Range Rules





Definition and Basic Rules

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Definition of Logarithm
log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x
Logarithm is the inverse operation of exponentiation
Logarithm of One
log⁡b(1)=0\log_b(1) = 0
The logarithm of 1 to any base equals 0
Logarithm of Base
log⁡b(b)=1\log_b(b) = 1
The logarithm of the base to itself equals 1
Logarithm of Base Power
log⁡b(bx)=x\log_b(b^x) = x
Logarithm and exponentiation with same base cancel out
Base to Logarithm Power
blog⁡b(x)=xb^{\log_b(x)} = x
Base raised to its own logarithm equals the argument

Fundamental Properties

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Product Rule
log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
Logarithm of a product equals sum of logarithms
Quotient Rule
log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)
Logarithm of a quotient equals difference of logarithms
Power Rule
log⁡b(xn)=n⋅log⁡b(x)\log_b(x^n) = n \cdot \log_b(x)
Logarithm of a power brings the exponent as a coefficient
Root Rule
log⁡b(xn)=1n⋅log⁡b(x)\log_b(\sqrt[n]{x}) = \frac{1}{n} \cdot \log_b(x)
Logarithm of a root becomes a fractional coefficient

Base Conversion Rules

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Change of Base Formula
log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}
Convert logarithm to any other base using division
Natural Logarithm Conversion
log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)}
Convert any logarithm using natural logarithms
Common Logarithm Conversion
log⁡b(x)=log⁡(x)log⁡(b)\log_b(x) = \frac{\log(x)}{\log(b)}
Convert any logarithm using common logarithms
Base Reciprocal Rule
log⁡1/b(x)=−log⁡b(x)\log_{1/b}(x) = -\log_b(x)
Logarithm with reciprocal base changes sign

Natural Logarithm Properties

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Natural Logarithm of e
ln⁡(e)=1\ln(e) = 1
Natural logarithm of e equals 1
Natural Logarithm of 1
ln⁡(1)=0\ln(1) = 0
Natural logarithm of 1 equals 0
Natural Logarithm of e Power
ln⁡(ex)=x\ln(e^x) = x
Natural logarithm and e exponent cancel out
e to Natural Logarithm Power
eln⁡(x)=xe^{\ln(x)} = x
e raised to natural logarithm equals the argument

Common Logarithm Properties

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Common Logarithm of 10
log⁡(10)=1\log(10) = 1
Common logarithm of 10 equals 1
Common Logarithm of 1
log⁡(1)=0\log(1) = 0
Common logarithm of 1 equals 0
Common Logarithm of 10 Power
log⁡(10x)=x\log(10^x) = x
Common logarithm and base 10 exponent cancel out
10 to Common Logarithm Power
10log⁡(x)=x10^{\log(x)} = x
10 raised to common logarithm equals the argument

Advanced Logarithmic Rules

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Reciprocal Rule
log⁡b(1x)=−log⁡b(x)\log_b\left(\frac{1}{x}\right) = -\log_b(x)
Logarithm of a reciprocal equals negative logarithm
Logarithm Equality
If log⁡b(x)=log⁡b(y)\log_b(x) = \log_b(y), then x=yx = y
Equal logarithms with same base have equal arguments
Exponential Equation Solver
If ax=bya^x = b^y, then x⋅log⁡(a)=y⋅log⁡(b)x \cdot \log(a) = y \cdot \log(b)
Solve exponential equations using logarithms
Compound Logarithm
log⁡b(log⁡c(x))\log_b(\log_c(x)) requires x>1x > 1 when c>1c > 1
Nested logarithms have restricted domains

Domain and Range Rules

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Domain Restriction
log⁡b(x)\log_b(x) is defined when x>0x > 0, b>0b > 0, b≠1b \neq 1
Logarithm argument must be positive, base positive and not 1
Sign Rule for Large Arguments
log⁡b(x)>0\log_b(x) > 0 when x>1x > 1 (for b>1b > 1)
Logarithm is positive when argument exceeds 1
Sign Rule for Small Arguments
log⁡b(x)<0\log_b(x) < 0 when 0<x<10 < x < 1 (for b>1b > 1)
Logarithm is negative when argument is between 0 and 1
Range Property
log⁡b(x)∈(−∞,∞)\log_b(x) \in (-\infty, \infty) for valid xx
Logarithm function has range of all real numbers