<p>For&nbsp;\(a&gt;0\), \(a\ne1\), the functions \(a^x\) and \(\log_a x\) are inverses. Their graphs reflect in \(y=x\). The special base \(e\) gives&nbsp;\(e^x\) and \(\ln x\).</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Function</strong></th><th style="text-align: center; --cell-justify: center"><strong>Domain; range</strong></th><th style="text-align: center; --cell-justify: center"><strong>Key features</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(a^x\)</td><td style="text-align: center; --cell-justify: center">\(\mathbb R;\ (0,\infty)\)</td><td style="text-align: center; --cell-justify: center">\((0,1)\); asymptote \(y=0\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\log_a x\)</td><td style="text-align: center; --cell-justify: center">\((0,\infty);\ \mathbb R\)</td><td style="text-align: center; --cell-justify: center">\((1,0)\); asymptote \(x=0\)</td></tr></tbody></table></div></div>
<p>Both increase if&nbsp;\(a&gt;1\) and decrease if&nbsp;\(0&lt;a&lt;1\). Also \(a^x=e^{x\ln a}\) and \(\log_a(a^x)=x\) for every real \(x\).</p>
<p class="paragraph-header">Example</p>
<p>\(y=\ln(x-2)\) has domain \(x&gt;2\), range \(\mathbb R\), asymptote \(x=2\), intercept \((3,0)\), and no \(y\)-intercept.</p>
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For \(a>0\), \(a\ne1\), the functions \(a^x\) and \(\log_a x\) are inverses. Their graphs reflect in \(y=x\). The special base \(e\) gives \(e^x\) and \(\ln x\).

Both increase if \(a>1\) and decrease if \(0<a<1\). Also \(a^x=e^{x\ln a}\) and \(\log_a(a^x)=x\) for every real \(x\).

\(y=\ln(x-2)\) has domain \(x>2\), range \(\mathbb R\), asymptote \(x=2\), intercept \((3,0)\), and no \(y\)-intercept.

Exponential and Logarithmic Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Exponential and logarithmic graphs

Exponential and logarithmic graphs

Function	Domain; range	Key features
\(a^x\)	\(\mathbb R;\ (0,\infty)\)	\((0,1)\); asymptote \(y=0\)
\(\log_a x\)	\((0,\infty);\ \mathbb R\)	\((1,0)\); asymptote \(x=0\)