Logarithm

algebra real-analysis logarithms

Definition

Logarithm

The logarithm is the inverse operation of exponentiation:

$$x={\log }_b(a):⟺b^x=a$$

Sometimes the basis is omitted to abbreviate the notation. Normally, this means that the basis is 10. However, depending on the context, this might not apply universally.

Rules

Product as Sum

$${\log }_a(x\cdot y)={\log }_a(x)+{\log }_a(y)$$

Fraction as Difference

$${\log }_c\left(\frac{a}{b}\right)={\log }_c(a)-{\log }_c(b)$$

Fraction as Difference

Fix $b>0$, $b\ne 1$, and $c>0$, $c\ne 1$. Define

$$f(a)={\log }_c\left(\frac{a}{b}\right),g(a)={\log }_c(a).$$

Then

$$f^{\prime }(a)=\frac{1}{\left(\frac{a}{b}\right)\ln c}\cdot \frac{1}{b}=\frac{1}{a\ln c}=g^{\prime }(a).$$

By the constant difference theorem, there exists a constant $k\in \mathbb{R}$ such that

$${\log }_c\left(\frac{a}{b}\right)={\log }_c(a)+k.$$

Evaluate at $a=b$:

$$0={\log }_c(1)={\log }_c\left(\frac{b}{b}\right)={\log }_c(b)+k,$$

so $k=-{\log }_c(b)$. Substituting back gives

$${\log }_c\left(\frac{a}{b}\right)={\log }_c(a)-{\log }_c(b).$$

Factor as Exponent

$$a\cdot {\log }_b(c)={\log }_b(c^a)$$

Root as Fraction

$${\log }_b(\sqrt[y]{x})=\frac{1}{y}\cdot {\log }_b(x)$$

Continuity

Continuity of Logarithm

The logarithmic function is continuous on its domain $(0,\infty )$.