\end{tabularx}
\vfill
+\vtop to 5cm {
+ \flushbottom
+ \subsubsection*{Index identities}
+ \begin{align*}
+ b^{m+n} &= b^m \cdot b^n \\
+ (b^m)^n &= b^{m \cdot n} \\
+ (b \cdot c)^n &= b^n \cdot c^n \\
+ {a^m \div a^n} &= {a^{m-n}}
+ \end{align*}
+}
+
\subsection*{Antiderivatives}
\vspace{1em}
Note \(\sin^{-1} \left(\dfrac{x}{a}\right) + \cos^{-1} \left(\dfrac{x}{a}\right)\) is constant \(\forall \> x \in (-a, a)\)
+
+\vfill
+\vtop to 5cm {
+ \flushbottom
+ \subsubsection*{Logarithmic identities}
+ \begin{align*}
+ \log_b (xy) &= \log_b x + \log_b y \\
+ \log_b\left(\frac{x}{y}\right) &= \log_b(x) - \log_b(y) \\
+ \log_b x^n &= n \log_b x \\
+ \log_b y^{x^n} &= x^n \log_b y
+ \end{align*}
+}