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Next: <a href="Complex-Arithmetic.html" accesskey="n" rel="next">Complex Arithmetic</a>, Up: <a href="Arithmetic.html" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<h3 class="section" id="Exponents-and-Logarithms-1"><span>17.1 Exponents and Logarithms<a class="copiable-link" href="#Exponents-and-Logarithms-1"> ¶</a></span></h3>
<a class="anchor" id="XREFexp"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-exp"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">exp</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-exp"> ¶</a></span></dt>
<dd><p>Compute
<code class="code">e^x</code>
for each element of <var class="var">x</var>.
</p>
<p>To compute the matrix exponential, see <a class="pxref" href="Linear-Algebra.html">Linear Algebra</a>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlog">log</a>.
</p></dd></dl>
<a class="anchor" id="XREFexpm1"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-expm1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">expm1</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-expm1"> ¶</a></span></dt>
<dd><p>Compute
<code class="code">exp (<var class="var">x</var>) - 1</code>
accurately in the neighborhood of zero.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFexp">exp</a>.
</p></dd></dl>
<a class="anchor" id="XREFlog"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-log"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">log</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-log"> ¶</a></span></dt>
<dd><p>Compute the natural logarithm,
<code class="code">ln (<var class="var">x</var>)</code>,
for each element of <var class="var">x</var>.
</p>
<p>To compute the matrix logarithm, see <a class="pxref" href="Linear-Algebra.html">Linear Algebra</a>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFexp">exp</a>, <a class="ref" href="#XREFlog1p">log1p</a>, <a class="ref" href="#XREFlog2">log2</a>, <a class="ref" href="#XREFlog10">log10</a>, <a class="ref" href="Special-Utility-Matrices.html#XREFlogspace">logspace</a>.
</p></dd></dl>
<a class="anchor" id="XREFreallog"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-reallog"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">reallog</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-reallog"> ¶</a></span></dt>
<dd><p>Return the real-valued natural logarithm of each element of <var class="var">x</var>.
</p>
<p>If any element results in a complex return value <code class="code">reallog</code> aborts and
issues an error.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlog">log</a>, <a class="ref" href="#XREFrealpow">realpow</a>, <a class="ref" href="#XREFrealsqrt">realsqrt</a>.
</p></dd></dl>
<a class="anchor" id="XREFlog1p"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-log1p"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">log1p</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-log1p"> ¶</a></span></dt>
<dd><p>Compute
<code class="code">log (1 + <var class="var">x</var>)</code>
accurately in the neighborhood of zero.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlog">log</a>, <a class="ref" href="#XREFexp">exp</a>, <a class="ref" href="#XREFexpm1">expm1</a>.
</p></dd></dl>
<a class="anchor" id="XREFlog10"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-log10"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">log10</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-log10"> ¶</a></span></dt>
<dd><p>Compute the base-10 logarithm of each element of <var class="var">x</var>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlog">log</a>, <a class="ref" href="#XREFlog2">log2</a>, <a class="ref" href="Special-Utility-Matrices.html#XREFlogspace">logspace</a>, <a class="ref" href="#XREFexp">exp</a>.
</p></dd></dl>
<a class="anchor" id="XREFlog2"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-log2"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">log2</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-log2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-log2-1"><span><code class="def-type">[<var class="var">f</var>, <var class="var">e</var>] =</code> <strong class="def-name">log2</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-log2-1"> ¶</a></span></dt>
<dd><p>Compute the base-2 logarithm of each element of <var class="var">x</var>.
</p>
<p>If called with one output, compute the base-2 logarithm such that
<code class="code">2^<var class="var">y</var> = <var class="var">x</var></code>.
</p>
<p>If called with two output arguments, split <var class="var">x</var> into binary mantissa
(<var class="var">f</var>) and exponent (<var class="var">e</var>) such that
<code class="code"><var class="var">x</var> = <var class="var">f</var> * 2^<var class="var">e</var></code>
where
<code class="code">1/2 <= abs (<var class="var">f</var>) < 1</code><!-- /@w -->
and <var class="var">e</var> is an integer. If
<code class="code">x = 0</code><!-- /@w -->, <code class="code">f = e = 0</code><!-- /@w -->.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFpow2">pow2</a>, <a class="ref" href="#XREFlog">log</a>, <a class="ref" href="#XREFlog10">log10</a>, <a class="ref" href="#XREFexp">exp</a>.
</p></dd></dl>
<a class="anchor" id="XREFpow2"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-pow2"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">pow2</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-pow2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-pow2-1"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">pow2</strong> <code class="def-code-arguments">(<var class="var">f</var>, <var class="var">e</var>)</code><a class="copiable-link" href="#index-pow2-1"> ¶</a></span></dt>
<dd><p>With one input argument, compute
y = 2 .^ x
for each element of <var class="var">x</var>.
</p>
<p>With two input arguments, return
y = f .* (2 .^ e).
where for complex inputs only the real part of both inputs is regarded
and from <var class="var">e</var> only the real integer part. This calling form corresponds
to C/C++ standard function <code class="code">ldexp()</code>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFlog2">log2</a>, <a class="ref" href="#XREFnextpow2">nextpow2</a>, <a class="ref" href="Arithmetic-Ops.html#XREFpower">power</a>.
</p></dd></dl>
<a class="anchor" id="XREFnextpow2"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-nextpow2"><span><code class="def-type"><var class="var">n</var> =</code> <strong class="def-name">nextpow2</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-nextpow2"> ¶</a></span></dt>
<dd><p>Compute the exponent of the next power of two not smaller than the input.
</p>
<p>For each element in the input array <var class="var">x</var>, return the smallest integer
<var class="var">n</var> such that
<code class="code">2^<var class="var">n</var> ≥ abs (<var class="var">x</var>)</code>.
For input elements equal to zero, return zero.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFpow2">pow2</a>, <a class="ref" href="#XREFlog2">log2</a>.
</p></dd></dl>
<a class="anchor" id="XREFrealpow"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-realpow"><span><code class="def-type"><var class="var">z</var> =</code> <strong class="def-name">realpow</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-realpow"> ¶</a></span></dt>
<dd><p>Compute the real-valued, element-by-element power operator.
</p>
<p>This is equivalent to <code class="code"><var class="var">x</var> .^ <var class="var">y</var></code><!-- /@w -->, except that
<code class="code">realpow</code> reports an error if any return value is complex.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Arithmetic-Ops.html#XREFpower">power</a>, <a class="ref" href="#XREFreallog">reallog</a>, <a class="ref" href="#XREFrealsqrt">realsqrt</a>.
</p></dd></dl>
<a class="anchor" id="XREFsqrt"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-sqrt"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">sqrt</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-sqrt"> ¶</a></span></dt>
<dd><p>Compute the square root of each element of <var class="var">x</var>.
</p>
<p>If <var class="var">x</var> is negative, a complex result is returned.
</p>
<p>To compute the matrix square root, see <a class="pxref" href="Linear-Algebra.html">Linear Algebra</a>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrealsqrt">realsqrt</a>, <a class="ref" href="#XREFnthroot">nthroot</a>.
</p></dd></dl>
<a class="anchor" id="XREFrealsqrt"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-realsqrt"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">realsqrt</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-realsqrt"> ¶</a></span></dt>
<dd><p>Return the real-valued square root of each element of <var class="var">x</var>.
</p>
<p>If any element results in a complex return value <code class="code">realsqrt</code> aborts and
issues an error.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFsqrt">sqrt</a>, <a class="ref" href="#XREFrealpow">realpow</a>, <a class="ref" href="#XREFreallog">reallog</a>.
</p></dd></dl>
<a class="anchor" id="XREFcbrt"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-cbrt"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">cbrt</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-cbrt"> ¶</a></span></dt>
<dd><p>Compute the real-valued cube root of each element of <var class="var">x</var>.
</p>
<p>Unlike <code class="code"><var class="var">x</var>^(1/3)</code>, the result will be negative if <var class="var">x</var> is
negative.
</p>
<p>If any element of <var class="var">x</var> is complex, <code class="code">cbrt</code> aborts with an error.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFnthroot">nthroot</a>.
</p></dd></dl>
<a class="anchor" id="XREFnthroot"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-nthroot"><span><code class="def-type"><var class="var">y</var> =</code> <strong class="def-name">nthroot</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">n</var>)</code><a class="copiable-link" href="#index-nthroot"> ¶</a></span></dt>
<dd>
<p>Compute the real (non-complex) <var class="var">n</var>-th root of <var class="var">x</var>.
</p>
<p><var class="var">x</var> must have all real entries and <var class="var">n</var> must be a scalar.
If <var class="var">n</var> is an even integer and <var class="var">x</var> has negative entries then
<code class="code">nthroot</code> aborts and issues an error.
</p>
<p>Example:
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">nthroot (-1, 3)
⇒ -1
(-1) ^ (1 / 3)
⇒ 0.50000 - 0.86603i
</pre></div></div>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFrealsqrt">realsqrt</a>, <a class="ref" href="#XREFsqrt">sqrt</a>, <a class="ref" href="#XREFcbrt">cbrt</a>.
</p></dd></dl>
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