Systems of exponential equations

A typical system has equations like $a^{f(x,y)}$ and $b^{g(x,y)}$ . The goal is to simplify powers step by step until you solve a simple system (often linear in $x,y$ ). Common moves: • reduce a cluster to one base, as in a single equation: $8^x=(2^3)^x=2^{3x}$ , $9^y=3^{2y}$ ; • obtain $A^{\text{linear}}=\text{const}$ , then compare exponents only ( $A>0$ , $A\neq 1$ ); • solve the resulting linear pair by substitution or elimination. Each pair $(x,y)$ must be plugged back into the original exponentials — errors come from extraneous solutions or from silently requiring positive powers only.

First — one base per equation (or a clean split via bases 2, 3, 5)

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