Binary logarithm

==Definition and properties==

==Definition and properties==

The binary logarithm function may be defined as the [[inverse function]] to the [[power of two]] function, which is a strictly increasing function over the positive [[real number]]s and therefore has a unique inverse.<ref>{{citation|title=Introduction to Mathematics for Life Scientists|publisher=Springer|year=2012|first=E.|last=Batschelet|isbn=978-3-642-96080-2|url=https://books.google.com/books?id=vbT0CAAAQBAJ&pg=PA128|page=128}}.</ref>

The binary logarithm function may be defined as the [[inverse function]] to the [[power of two]] function, which is a strictly increasing function over the positive [[real number]]s and therefore has a unique inverse.<ref>{{citation|title=Introduction to Mathematics for Life Scientists|publisher=Springer|year=2012|first=E.|last=Batschelet|isbn=978-3-642-96080-2|url=https://books.google.com/books?id=vbT0CAAAQBAJ&pg=PA128|page=128}}.</ref>

Alternatively, it may be defined as {{math|(ln ''n'')/(ln 2)}}, where {{math|ln}} is the [[natural logarithm]], defined in any of its standard ways. Using the [[complex logarithm]] in this definition allows the binary logarithm to be extended to the [[complex number]]s.<ref>For instance, [[Microsoft Excel]] provides the <code>IMLOG2</code> function for complex binary logarithms: see {{citation|title=Excel Scientific and Engineering Cookbook|first=David M.|last=Bourg|publisher=O'Reilly Media|year=2006|isbn=978-0-596-55317-3|page=232|url=https://books.google.com/books?id=uKctiVg2dyIC&pg=PT248}}.</ref>

Alternatively, it may be defined as {{math|ln ''n''/ln 2}}, where {{math|ln}} is the [[natural logarithm]], defined in any of its standard ways. Using the [[complex logarithm]] in this definition allows the binary logarithm to be extended to the [[complex number]]s.<ref>For instance, [[Microsoft Excel]] provides the <code>IMLOG2</code> function for complex binary logarithms: see {{citation|title=Excel Scientific and Engineering Cookbook|first=David M.|last=Bourg|publisher=O'Reilly Media|year=2006|isbn=978-0-596-55317-3|page=232|url=https://books.google.com/books?id=uKctiVg2dyIC&pg=PT248}}.</ref>

As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation:<ref>{{citation|title=Algebra for College Students|first1=Bernard|last1=Kolman|first2=Arnold|last2=Shapiro|publisher=Academic Press|year=1982|isbn=978-1-4832-7121-7|url=https://books.google.com/books?id=i7vSBQAAQBAJ&pg=PA334|pages=334–335|contribution=11.4 Properties of Logarithms}}.</ref>

As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation:<ref>{{citation|title=Algebra for College Students|first1=Bernard|last1=Kolman|first2=Arnold|last2=Shapiro|publisher=Academic Press|year=1982|isbn=978-1-4832-7121-7|url=https://books.google.com/books?id=i7vSBQAAQBAJ&pg=PA334|pages=334–335|contribution=11.4 Properties of Logarithms}}.</ref>