Exponentials & Logarithms

Whenever an equation is built from different powers of a single exponential block or a single logarithm, set $u$ equal to that block. The equation collapses to an ordinary polynomial (usually a quadratic). Solve for $u$, then read off $x$.

Trigger: The variable appears inside powers of one exponential base, or as powers of a logarithm like $(\log x{)}^2$.

Instances:
(i) for exponentials, split additions in the exponent first ($a^{2x+1}=a(a^x{)}^2$) and reject any $u\le 0$ since $a^x>0$;
(ii) for logarithms, pull exponents out of arguments first ($\log (x^2)=2\log x$) before substituting $u=\log x$;
(iii) for bases like $\sqrt{2}$, setting $u=(\sqrt{2}{)}^x$ makes $2^x=u^2$.