TMUA Exponentials and Logarithms

TMUA does not include $e^x$ or $\ln x$ — the syllabus is restricted to base-$a$ exponentials and logarithms for real $a > 0$, $a \ne 1$. What does show up, in nearly every Paper 1, is an exponential equation that becomes a quadratic in disguise the moment you make the right substitution.

• Quadratic-in-disguise. Equations of the form $a^{2x} - p \cdot a^x + q = 0$ become a standard quadratic under $u = a^x$.
• Equation between logs. $\log_a M = \log_a N \Rightarrow M = N$, provided both $M, N > 0$. The positivity condition is part of the answer.
• Change of base. $\log_2 3 = 1 / \log_3 2$. Short identity chain, costly when forgotten.
• Index laws under pressure. $a^{x+y} = a^x \cdot a^y$ and $a^{xy} = (a^x)^y$.

The move. Whenever you see $a^{kx}$ and $a^{x}$ in the same equation, substitute $u = a^{x}$. The first term becomes $u^k$ and the equation is polynomial. Solve in $u$, then convert back through $u = a^x$, checking $u > 0$.