Exponents, Logarithms, and Roots Key concepts

This folder is about one operation and its two inverses. If

$$b^n=x,$$

then Exponentiation asks for $x$, Logarithm asks for $n$, and Root asks for $b$.

Core concepts

• Exponent laws are bookkeeping rules for repeated multiplication: $b^m{b}^n=b^{m+n}$, $(b^m{)}^n=b^{mn}$, and $b^{-n}=1/b^n$.
• A power function has variable base, $f(x)=x^n$; an exponential function has variable exponent, $f(x)=b^x$.
• Logarithms turn multiplication into addition: ${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$.
• Roots are fractional powers: $\sqrt[n]{x}=x^{1/n}$, with domain care for even roots of real numbers.
• The natural base $e$ is special because $\frac{d}{dx}{e}^x=e^x$ and $\frac{d}{dx}\ln x=1/x$.

Undergrad to postgraduate bridge

Exponentials diagonalise many linear rate laws: if $y^{\prime }=ky$, then $y(t)=C{e}^{kt}$. Logarithms invert scale: a multiplicative ratio becomes an additive difference, which is why they appear in entropy, likelihoods, decibels, pH, and cosmological magnitude scales such as Cosmology distance methods. Complex powers require choosing a branch of the logarithm, so $z^a=e^{a\text{Log}z}$ is multi-valued unless a branch is fixed.

Common pitfalls

• Treating exponentiation as commutative: $2^3\ne 3^2$.
• Using log laws when bases differ.
• Forgetting that ${\log }_{b}x$ requires $b>0$, $b\ne 1$, and $x>0$ over the reals.
• Assuming $\sqrt{x^2}=x$; over the reals it is $|x|$.

Local notes

• Exponents, Logarithms, and Roots index
• Exponents, Logarithms, and Roots
• Exponentiation
• Logarithm
• Root
• Growth and decay