Indices

<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">\(y=a^x\qquad a&gt;0,\ a\ne1\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Value</strong></th><th style="text-align: center; --cell-justify: center"><strong>Result</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(a^0\)</td><td style="text-align: center; --cell-justify: center">\(1\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(a^1\)</td><td style="text-align: center; --cell-justify: center">\(a\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(a^x\)</td><td style="text-align: center; --cell-justify: center">\(&gt;0\) for every real \(x\)</td></tr></tbody></table></div></div>
<p class="paragraph-header">The natural base</p>
<p>\(e\approx2.718\). The index laws also apply to powers of \(e\).</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 40.7407%;"><col style="width: 59.2593%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Law applied to base \(\mathrm{e}\)</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">\(\mathrm{e}^m\,\mathrm{e}^n=\mathrm{e}^{m+n}\)</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">\((\mathrm{e}^m)^n=\mathrm{e}^{mn}\)</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">\(\mathrm{e}^{-m}=\dfrac{1}{\mathrm{e}^m}\)</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">\(\mathrm{e}^{0}=1\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>An exponential with a positive base is never zero or negative.</p>

\(e\approx2.718\). The index laws also apply to powers of \(e\).

\(\mathrm{e}^m\,\mathrm{e}^n=\mathrm{e}^{m+n}\)

\(\mathrm{e}^{x}\,\mathrm{e}^{2x}=\mathrm{e}^{3x}\)

\(\mathrm{e}^{-m}=\dfrac{1}{\mathrm{e}^m}\)

\(\mathrm{e}^{-x}=\dfrac{1}{\mathrm{e}^x}\)

An exponential with a positive base is never zero or negative.

The Exponential Function and the Base e

The Exponential Function and the Base e

Law applied to base \(\mathrm{e}\)	Example
\(\mathrm{e}^m\,\mathrm{e}^n=\mathrm{e}^{m+n}\)
\((\mathrm{e}^m)^n=\mathrm{e}^{mn}\)
\(\mathrm{e}^{-m}=\dfrac{1}{\mathrm{e}^m}\)
\(\mathrm{e}^{0}=1\)

Value	Result
\(a^0\)	\(1\)
\(a^1\)	\(a\)
\(a^x\)	\(>0\) for every real \(x\)