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