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| In [[mathematics]], there are several '''[[logarithm]]ic [[identity (mathematics)|identities]]'''.
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| == Algebraic identities or laws ==
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| === Trivial identities ===
| | This is the most useful way of obtain high quality groups that are right for you personally. It is also the best way for some players to boost their activities. The cost savings you"d accomplish in-the other techniques will be gone, but you will have good quality clone clubs that fit your move. Do leading PGA Tour players use off-the-shelf common clubs? Not in your life. Even though they endorse particular manufacturers and us... <br><br>Less Risk- More Cost- Search for a Certified Team Maker/Fitter <br><br>This is the most useful approach to get good quality groups which can be right for you personally. It is also the simplest way for most players to enhance their games. The price savings you"d accomplish in the other techniques will be gone, but you will have top quality clone golf clubs that fit your swing. Do prime PGA Tour players use off-the-shelf regular clubs? Not in your life. Despite the fact that they recommend specific brands and use clubs from their sponsors, those clubs have already been modified and tailored for every player. To get one more viewpoint, we know you check out: [http://aibwf.info/blogs/how-to-get-google-adsense-to-target-a-particular-section-of-a-net-page/ copyright]. In the event the most useful people on earth have to have their groups customized for his or her personal swings, wouldnt it benefit you as well, whose move is less perfect? If you have a set of clubs produced by a professional club maker/fitter who"s authorized by PCS (the Professional Clubmakers Society- look for a PCS Class A clubmaker) or the GCA (Golf Clubmakers Assoc.), you"ll receive a set of clubs that will get everything possible out of your game. If this process is beyond the way of your budget, it"s possible to perform a club installing on-line. That s-olution would be your best guess, If you can locate a professional licensed clubmaker in one of the 2 professional companies mentioned above who offers on-line clubmaking and installation services. To recognize this type of person, look on the web site of the PCA. <br><br>Some Principles Before Getting Clubs <br><br>Your professional team producer can obtain a specific amount of information before selecting the parts to make your [http://www.Bbc.co.uk/search/?q=clone+golf clone golf] clubs, and you should know these records even when you obtain a set of standard, off-the-shelf clubs. Hell measure your swing speed with both a driver and five iron. Hell gauge the distance from the ground to the top of our arm to ascertain the correct length on your clubs (most common clubs are too much time). We learned about [http://artofplayboy.info/blogs/making-your-backlinks-depend-i/ Making Your Backlinks Depend I | Art of Playboy] by browsing Google. He will discover your move and determine the proper loft on your driver and other clubs (many golfers use a driver with not enough loft). Hell make necessary adjustments in the club face settings and determine in the event that you usually slice or fade the-ball. Hell make sure you have the proper lie direction in your irons, which could substantially improve accuracy. Get further on our related article directory - Navigate to this web page: [http://annieswebdesign.info/blogs/how-sick-people-progress-rest/ linklicious plugin]. In all there are more than two-dozen adjustments a club [http://imageshack.us/photos/machine machine] can differ to boost groups for your swing, although of course some are more essential than others. <br><br>Conclusion <br><br>You can approach acquiring some clone golf clubs in three ways. You can just find a site that offers affordable clones, you can purchase quality components and assemble them yourself, or you can have the best set easy for your game and go to a certified professional team maker/fitter. Even though you cant pay the latter method, you should at the very least know your swing speeds to be able to select proper shafts, and you should get groups that are the proper length and proper attic for the swing and size. In addition, I"d recommend that anyone looking to choose new pair of golf clubs read Tom Wishons book, The Search for the Perfect Golf Club..<br><br>If you have any queries about where and how to use [http://www.purevolume.com/listeners/amuckyears6132 secondary health insurance], you can make contact with us at our web site. |
| {| cellpadding=3
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| | <math> \log_b(1) = 0 \!\, </math> || because || <math> b^0 = 1\!\, </math>, given that ''b>0''
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| | <math> \log_b(b) = 1 \!\, </math> || because || <math> b^1 = b\!\, </math>
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| |}
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| Note that log<sub>''b''</sub>(0) is undefined because there is no number ''x'' such that ''b''<sup>''x''</sup> = 0. In fact, there is a [[vertical asymptote]] on the graph of log<sub>''b''</sub>(''x'') at ''x'' = 0.
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| === Canceling exponentials ===
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| Logarithms and exponentials (antilogarithms) with the same base cancel each other. This is true because logarithms and exponentials are inverse operations (just like multiplication and division or addition and subtraction).
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| : <math> b^{\log_b(x)} = x\text{ because }\operatorname{antilog}_b(\log_b(x)) = x \, </math>
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| : <math> \log_b(b^x) = x\text{ because }\log_b(\operatorname{antilog}_b(x)) = x \, </math>
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| Both of the above are derived from the following two equations that define a logarithm:-
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| : <math> b^c = x\text{, }\log_b(x) = c \, </math>
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| Substituting c in the left equation gives b<big><sup>log<sub>b</sub>(x)</sup></big> = x, and substituting x in the right gives log<sub>b</sub>(b<big><sup>c</sup></big>) = c. Finally, replace c by x.
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| === Using simpler operations ===
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| Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. The first three operations below assume {{nowrap begin}}x = b<big><sup>c</sup></big>{{nowrap end}}, and/or {{nowrap begin}}y = b<big><sup>d</sup></big>{{nowrap end}} so that {{nowrap begin}}log<sub>b</sub>(x) = c{{nowrap end}} and {{nowrap begin}}log<sub>b</sub>(y) = d{{nowrap end}}. Derivations also use the log definitions {{nowrap begin}}x = b<big><sup>log<sub>b</sub>(x)</sup></big>{{nowrap end}} and {{nowrap begin}}x = log<sub>b</sub>(b<sup>x</sup>){{nowrap end}}.
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| {| cellpadding=3
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| | <math> \log_b(xy) = \log_b(x) + \log_b(y) \!\, </math> || because || <math> b^c \cdot b^d = b^{c + d} \!\, </math>
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| | <math> \log_b\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right) = \log_b(x) - \log_b(y) </math> || because || <math> b^{c-d} = \tfrac{b^c}{b^d} </math>
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| | <math> \log_b(x^d) = d \log_b(x) \!\, </math> || because || <math> (b^c)^d = b^{cd} \!\, </math>
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| | <math> \log_b\!\left(\!\sqrt[y]{x}\right) = \begin{matrix}\frac{\log_b(x)}{y}\end{matrix} </math> || because || <math> \sqrt[y]{x} = x^{1/y} </math>
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| | <math> x^{\log_b(y)} = y^{\log_b(x)} \!\, </math> || because || <math> x^{\log_b(y)} = b^{\log_b(x) \log_b(y)} = b^{\log_b(y) \log_b(x)} = y^{\log_b(x)} \!\, </math>
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| | <math> c\log_b(x)+d\log_b(y) = \log_b(x^c y^d) \!\, </math> || because || <math> \log_b(x^c y^d) = \log_b(x^c) + \log_b(y^d) \!\, </math>
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| Where <math>b</math>, <math>x</math>, and <math>y</math> are positive real numbers and <math>b \ne 1</math>. Both <math>c</math> and <math>d</math> are real numbers.
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| The laws result from canceling exponentials and appropriate law of indices. Starting with the first law:
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| <math>xy = b^{\log_b(x)} b^{\log_b(y)} = b^{\log_b(x) + \log_b(y)} \Rightarrow \log_b(xy) = \log_b(b^{\log_b(x) + \log_b(y)}) = \log_b(x) + \log_b(y)</math>
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| The law for powers exploits another of the laws of indices:
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| <math>x^y = (b^{\log_b(x)})^y = b^{y \log_b(x)} \Rightarrow \log_b(x^y) = y \log_b(x)</math>
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| The law relating to quotients then follows:
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| <math>\log_b \bigg(\frac{x}{y}\bigg) = \log_b(x y^{-1}) = \log_b(x) + \log_b(y^{-1}) = \log_b(x) - \log_b(y)</math>
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| Similarly, the root law is derived by rewriting the root as a reciprocal power:
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| <math>\log_b(\sqrt[y]x) = \log_b(x^{\frac{1}{y}}) = \frac{1}{y}\log_b(x)</math>
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| === Changing the base ===
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| :<math>\log_b a = {\log_d a \over \log_d b}</math>
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| This identity is useful to evaluate logarithms on calculators. For instance, most calculators have buttons for [[Natural logarithm|ln]] and for log<sub>10</sub>, but not for log<sub>2</sub>. To find log<sub>2</sub>(3), one could calculate log<sub>10</sub>(3) / log<sub>10</sub>(2) (or ln(3)/ln(2), which yields the same result).
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| ==== Proof ====
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| :Let <math>c=\log_b a</math>.
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| :Then <math>b^c=a</math>.
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| :Take <math>\log_d</math> on both sides: <math>\log_d b^c=\log_d a</math>
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| :Simplify and solve for <math>c</math>: <math> c\log_d b=\log_d a</math>
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| :<math>c=\frac{\log_d a}{\log_d b}</math>
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| :Since <math>c=\log_b a</math>, then <math>\log_b a=\frac{\log_d a}{\log_d b}</math>
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| This formula has several consequences:
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| :<math> \log_b a = \frac {1} {\log_a b} </math>
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| :<math> \log_{b^n} a = {{\log_b a} \over n} </math>
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| :<math> b^{\log_a d} = d^{\log_a b} </math>
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| :<math>- \log_b a = \log_b \left({1 \over a}\right) = \log_{1 \over b} a</math>
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| <!-- extra blank space between two lines of "displayed" [[TeX]] for legibility -->
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| :<math> \log_{b_1}a_1 \,\cdots\, \log_{b_n}a_n | |
| = \log_{b_{\pi(1)}}a_1\, \cdots\, \log_{b_{\pi(n)}}a_n, \, </math>
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| where <math>\scriptstyle\pi\,</math> is any [[permutation]] of the subscripts 1, ..., ''n''. For example
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| :<math> \log_b w\cdot \log_a x\cdot \log_d c\cdot \log_d z
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| = \log_d w\cdot \log_b x\cdot \log_a c\cdot \log_d z. \, </math>
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| === Summation/subtraction ===
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| The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:
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| :<math>\log_b (a+c) = \log_b a + \log_b (1+b^{\log_b c - \log_b a})</math>
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| :<math>\log_b (a-c) = \log_b a + \log_b (1-b^{\log_b c - \log_b a})</math>
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| which gives the special cases:
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| :<math>\log_b (a+c) = \log_b a + \log_b \left(1+\frac{c}{a}\right)</math>
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| :<math>\log_b (a-c) = \log_b a + \log_b \left(1-\frac{c}{a}\right)</math>
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| Note that in practice <math>a</math> and <math>c</math> have to be switched on the right hand side of the equations if <math>c>a</math>. Also note that the subtraction identity is not defined if <math>a=c</math> since the logarithm of zero is not defined.
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| More generally:
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| :<math>\log _b \sum\limits_{i=0}^N a_i = \log_b a_0 + \log_b \left( 1+\sum\limits_{i=1}^N \frac{a_i}{a_0} \right) = \log _b a_0 + \log_b \left( 1+\sum\limits_{i=1}^N b^{\left( \log_b a_i - \log _b a_0 \right)} \right)</math>
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| where <math>a_0,\ldots ,a_N > 0</math>.
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| === Exponents ===
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| A useful identity involving exponents:
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| :<math> x^{\frac{\log(\log(x))}{\log(x)}} = \log(x) </math>
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| == Calculus identities ==
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| === [[Limit of a function|Limits]] ===
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| :<math>\lim_{x \to 0^+} \log_a x = -\infty \quad \mbox{if } a > 1</math>
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| :<math>\lim_{x \to 0^+} \log_a x = +\infty \quad \mbox{if } a < 1</math>
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| :<math>\lim_{x \to+\infty} \log_a x= +\infty \quad \mbox{if } a > 1</math> | |
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| :<math>\lim_{x \to+\infty} \log_a x= -\infty \quad \mbox{if } a < 1</math> | |
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| :<math>\lim_{x \to 0^+} x^b \log_a x = 0 \quad \mbox{if } b > 0</math>
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| :<math>\lim_{x \to+\infty} {1 \over x^b} \log_a x = 0 \quad \mbox{if } b > 0</math>
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| The last limit is often summarized as "logarithms grow more slowly than any power or root of ''x''".
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| === [[Derivative]]s of logarithmic functions ===
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| :<math>{d \over dx} \ln x = {1 \over x },</math>
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| :<math>{d \over dx} \log_b x = {1 \over x \ln b},</math>
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| Where <math>x > 0</math>, <math>b > 0</math>, and <math>b \ne 1</math>.
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| === Integral definition ===
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| :<math>\ln x = \int_1^x \frac {1}{t} dt </math>
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| === [[Integral]]s of logarithmic functions ===
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| : <math>\int \log_a x \, dx = x(\log_a x - \log_a e) + C</math>
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| To remember higher integrals, it's convenient to define:
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| :<math>x^{\left [n \right]} = x^{n}(\log(x) - H_n)</math>
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| Where <math>H_n</math> is the nth [[Harmonic number]].
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| :<math>x^{\left [ 0 \right ]} = \log x</math>
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| :<math>x^{\left [ 1 \right ]} = x \log(x) - x</math> | |
| :<math>x^{\left [ 2 \right ]} = x^2 \log(x) - \begin{matrix} \frac{3}{2} \end{matrix} \, x^2</math>
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| :<math>x^{\left [ 3 \right ]} = x^3 \log(x) - \begin{matrix} \frac{11}{6} \end{matrix} \, x^3</math>
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| Then,
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| :<math>\frac {d}{dx} \, x^{\left [ n \right ]} = n \, x^{\left [ n-1 \right ]}</math>
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| :<math>\int x^{\left [ n \right ]}\,dx = \frac {x^{\left [ n+1 \right ]}} {n+1} + C</math>
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| == Approximating large numbers ==
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| The identities of logarithms can be used to approximate large numbers. Note that log<sub>''b''</sub>(''a'') + log<sub>''b''</sub>(''c'') = log<sub>''b''</sub>(''ac''), where ''a'', ''b'', and ''c'' are arbitrary constants. Suppose that one wants to approximate the 44th [[Mersenne prime]], 2<sup>32,582,657</sup> − 1. To get the base-10 logarithm, we would multiply 32,582,657 by log<sub>10</sub>(2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 10<sup>9,808,357</sup> × 10<sup>0.09543</sup> ≈ 1.25 × 10<sup>9,808,357</sup>.
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| Similarly, factorials can be approximated by summing the logarithms of the terms.
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| == Complex logarithm identities ==
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| The [[complex logarithm]] is the [[complex number]] analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However a [[multivalued function]] can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a [[Riemann surface]]. A single valued version called the [[principal value]] of the logarithm can be defined which is discontinuous on the negative x axis and equals the multivalued version on a single [[branch cut]].
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| === Definitions ===
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| The convention will be used here that a capital first letter is used for the principal value of functions and the lower case version refers to the multivalued function. The single valued version of definitions and identities is always given first followed by a separate section for the multiple valued versions.
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| :ln(''r'') is the standard natural logarithm of the real number ''r''.
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| :Log(''z'') is the principal value of the complex logarithm function and has imaginary part in the range (-π, π].
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| :Arg(''z'') is the principal value of the [[Arg (mathematics)|arg]] function, its value is restricted to (-π, π]. It can be computed using Arg(''x''+''iy'')= [[atan2]](''y'', ''x'').
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| :<math>\operatorname{Log}(z) = \ln(|z|) + i \operatorname{Arg}(z)</math>
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| :<math>e^{\operatorname{Log}(z)} = z</math>
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| The multiple valued version of log(''z'') is a set but it is easier to write it without braces and using it in formulas follows obvious rules. | |
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| :log(''z'') is the set of complex numbers ''v'' which satisfy e<sup>''v''</sup> = ''z''
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| :arg(''z'') is the set of possible values of the [[Arg (mathematics)|arg]] function applied to ''z''.
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| When ''k'' is any integer:
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| :<math>\log(z) = \ln(|z|) + i \arg(z)</math>
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| :<math>\log(z) = \operatorname{Log}(z) + 2 \pi i k</math>
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| :<math>e^{\log(z)} = z</math>
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| === Constants ===
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| Principal value forms:
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| :<math>\operatorname{Log}(1) = 0</math>
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| :<math>\operatorname{Log}(e) = 1</math>
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| Multiple value forms, for any ''k'' an integer:
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| :<math>\log(1) = 0 + 2 \pi i k</math>
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| :<math>\log(e) = 1 + 2 \pi i k</math>
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| === Summation ===
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| Principal value forms:
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| :<math>\operatorname{Log}(z_1) + \operatorname{Log}(z_2) = \operatorname{Log}(z_1 z_2) \pmod {2 \pi i}</math>
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| :<math>\operatorname{Log}(z_1) - \operatorname{Log}(z_2) = \operatorname{Log}(z_1 / z_2) \pmod {2 \pi i}</math>
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| Multiple value forms:
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| :<math>\log(z_1) + \log(z_2) = \log(z_1 z_2)</math>
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| :<math>\log(z_1) - \log(z_2) = \log(z_1 / z_2)</math>
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| === Powers ===
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| A complex power of a complex number can have many possible values.
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| Principal value form:
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| :<math>{z_1}^{z_2} = e^{z_2 \operatorname{Log}(z_1)} </math>
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| :<math>\operatorname{Log}{\left({z_1}^{z_2}\right)} = z_2 \operatorname{Log}(z_1) \pmod {2 \pi i}</math>
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| Multiple value forms:
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| :<math>{z_1}^{z_2} = e^{z_2 \log(z_1)}</math>
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| Where ''k''<sub>1</sub>, ''k''<sub>2</sub> are any integers:
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| :<math>\log{\left({z_1}^{z_2}\right)} = z_2 \log(z_1) + 2 \pi i k_2</math>
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| :<math>\log{\left({z_1}^{z_2}\right)} = z_2 \operatorname{Log}(z_1) + z_2 2 \pi i k_1 + 2 \pi i k_2</math>
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| == See also ==
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| * [[List of trigonometric identities]]
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| * [[Exponential function]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{MathWorld|Logarithm|Logarithm}}
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| * [http://www.mathwords.com/l/logarithm.htm Logarithm] in Mathwords
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| | |
| [[Category:Logarithms]]
| |
| [[Category:Mathematical identities]]
| |
| [[Category:Articles containing proofs]]
| |
This is the most useful way of obtain high quality groups that are right for you personally. It is also the best way for some players to boost their activities. The cost savings you"d accomplish in-the other techniques will be gone, but you will have good quality clone clubs that fit your move. Do leading PGA Tour players use off-the-shelf common clubs? Not in your life. Even though they endorse particular manufacturers and us...
Less Risk- More Cost- Search for a Certified Team Maker/Fitter
This is the most useful approach to get good quality groups which can be right for you personally. It is also the simplest way for most players to enhance their games. The price savings you"d accomplish in the other techniques will be gone, but you will have top quality clone golf clubs that fit your swing. Do prime PGA Tour players use off-the-shelf regular clubs? Not in your life. Despite the fact that they recommend specific brands and use clubs from their sponsors, those clubs have already been modified and tailored for every player. To get one more viewpoint, we know you check out: copyright. In the event the most useful people on earth have to have their groups customized for his or her personal swings, wouldnt it benefit you as well, whose move is less perfect? If you have a set of clubs produced by a professional club maker/fitter who"s authorized by PCS (the Professional Clubmakers Society- look for a PCS Class A clubmaker) or the GCA (Golf Clubmakers Assoc.), you"ll receive a set of clubs that will get everything possible out of your game. If this process is beyond the way of your budget, it"s possible to perform a club installing on-line. That s-olution would be your best guess, If you can locate a professional licensed clubmaker in one of the 2 professional companies mentioned above who offers on-line clubmaking and installation services. To recognize this type of person, look on the web site of the PCA.
Some Principles Before Getting Clubs
Your professional team producer can obtain a specific amount of information before selecting the parts to make your clone golf clubs, and you should know these records even when you obtain a set of standard, off-the-shelf clubs. Hell measure your swing speed with both a driver and five iron. Hell gauge the distance from the ground to the top of our arm to ascertain the correct length on your clubs (most common clubs are too much time). We learned about Making Your Backlinks Depend I | Art of Playboy by browsing Google. He will discover your move and determine the proper loft on your driver and other clubs (many golfers use a driver with not enough loft). Hell make necessary adjustments in the club face settings and determine in the event that you usually slice or fade the-ball. Hell make sure you have the proper lie direction in your irons, which could substantially improve accuracy. Get further on our related article directory - Navigate to this web page: linklicious plugin. In all there are more than two-dozen adjustments a club machine can differ to boost groups for your swing, although of course some are more essential than others.
Conclusion
You can approach acquiring some clone golf clubs in three ways. You can just find a site that offers affordable clones, you can purchase quality components and assemble them yourself, or you can have the best set easy for your game and go to a certified professional team maker/fitter. Even though you cant pay the latter method, you should at the very least know your swing speeds to be able to select proper shafts, and you should get groups that are the proper length and proper attic for the swing and size. In addition, I"d recommend that anyone looking to choose new pair of golf clubs read Tom Wishons book, The Search for the Perfect Golf Club..
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