Editing Arithmetic (section)

=== Exponentiation and logarithm ===
{{main|Exponentiation|Logarithm}}

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Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in [[superscript]] right after the base. Examples are <math>2^4 = 16</math> and <math>3</math>^<math>3 = 27</math>. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in <math>2^4 = 2 \times 2 \times 2 \times 2</math>.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA117 117–118]}} | {{harvnb|Kay|2021|pp=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA27 27–28]}} }}</ref>{{efn|If the exponent is 0 then the result is 1, as in <math>7^0 = 1</math>. The only exception is <math>0^0</math>, which is not defined.<ref>{{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA120 120]}}</ref>}}

Roots are a special type of exponentiation using a fractional exponent. For example, the [[square root]] of a number is the same as raising the number to the power of <math>\tfrac{1}{2}</math> and the [[cube root]] of a number is the same as raising the number to the power of <math>\tfrac{1}{3}</math>. Examples are <math>\sqrt{4} = 4^{\frac{1}{2}} = 2</math> and <math>\sqrt[3]{27} = 27^{\frac{1}{3}} = 3</math>.<ref>{{multiref | {{harvnb|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA118 118]}} | {{harvnb|Klose|2014|p=[https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA105 105]}} }}</ref>

Logarithm is the inverse of exponentiation. The logarithm of a number <math>x</math> to the base <math>b</math> is the [[exponent]] to which <math>b</math> must be raised to produce <math>x</math>. For instance, since <math>1000 = 10^3</math>, the logarithm base 10 of 1000 is 3. The logarithm of <math>x</math> to base <math>b</math> is denoted as <math>\log_b (x)</math>, or without parentheses, <math>\log_b x</math>, or even without the explicit base, <math>\log x</math>, when the base can be understood from context. So, the previous example can be written <math>\log_{10} 1000 = 3</math>.<ref>{{multiref | {{harvnb|Kay|2021|pp=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA121 121–122]}} | {{harvnb|Rodda|Little|2015|p=[https://books.google.com/books?id=cb_dCgAAQBAJ&pg=PA7 7]}} }}</ref>

Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in <math>14^1 = 14</math>. However, exponentiation does not have a general identity element since 1 is not the neutral element for the base.<ref>{{multiref | {{harvnb|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA117 117]}} | {{harvnb|Mazzola|Milmeister|Weissmann|2004|p=[https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66 66]}} }}</ref> Exponentiation and logarithm are neither commutative nor associative.<ref>{{multiref | {{harvnb|Sally|Sally (Jr.)|2012|p=[https://books.google.com/books?id=Ntjq07-FA_IC&pg=PA3 3]}} | {{harvnb|Klose|2014|pp=[https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA107 107–108]}} }}</ref>