<p>For&nbsp;\(a&gt;0\) and \(a\ne1\):</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(a^u=a^v\quad\Longleftrightarrow\quad u=v\)</p></div></div>
<p class="paragraph-header">Methods</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 33.3333%;"><col style="width: 33.3333%;"><col style="width: 33.3333%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Form</strong></th><th style="text-align: center; --cell-justify: center"><strong>Method</strong></th><th style="text-align: center; --cell-justify: center"><strong>Example</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Powers of the same base</td><td style="text-align: center; --cell-justify: center">Equate indices.</td><td style="text-align: center; --cell-justify: center">\(2^{3x-1}=8\Rightarrow3x-1=3\Rightarrow x=\frac43\)</td></tr><tr><td style="text-align: center; --cell-justify: center">Quadratic in \(a^x\)</td><td style="text-align: center; --cell-justify: center">Let&nbsp;\(u=a^x&gt;0\).</td><td style="text-align: center; --cell-justify: center">Solve for u, reject non-positive values, then find x.</td></tr></tbody></table></div></div>
<p class="paragraph-header">Useful substitutions</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 37.0370%;"><col style="width: 35.1852%;"><col style="width: 27.7778%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Equation</strong></th><th style="text-align: center; --cell-justify: center"><strong>Substitution</strong></th><th style="text-align: center; --cell-justify: center"><strong>Because</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(9^x-4(3^x)+3=0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(\mathrm{e}^{2x}-7\mathrm{e}^{x}+10=0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(\mathrm{e}^{4x}-3\mathrm{e}^{2x}-4=0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(25^x=4-3(25^{-x})\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Factor out shifted powers first, e.g. \(2^{x+1}=2(2^x)\).</p>

\(a^u=a^v\quad\Longleftrightarrow\quad u=v\)

\(2^{3x-1}=8\Rightarrow3x-1=3\Rightarrow x=\frac43\)

Solve for u, reject non-positive values, then find x.

\(u=\mathrm{e}^{2x},\ u^2=\mathrm{e}^{4x}\)

Factor out shifted powers first, e.g. \(2^{x+1}=2(2^x)\).

Solving Exponential Equations

Solving Exponential Equations

Form	Method	Example
Powers of the same base	Equate indices.	\(2^{3x-1}=8\Rightarrow3x-1=3\Rightarrow x=\frac43\)
Quadratic in \(a^x\)	Let \(u=a^x>0\).	Solve for u, reject non-positive values, then find x.

Equation	Substitution	Because
\(9^x-4(3^x)+3=0\)
\(\mathrm{e}^{2x}-7\mathrm{e}^{x}+10=0\)
\(\mathrm{e}^{4x}-3\mathrm{e}^{2x}-4=0\)
\(25^x=4-3(25^{-x})\)