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In [[mathematics]], there are several '''[[logarithm]]ic [[identity (mathematics)|identities]]'''.


== Algebraic identities or laws ==


=== Trivial identities ===
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{| cellpadding=3
| <math> \log_b(1) = 0 \!\, </math> || because || <math> b^0 = 1\!\, </math>, given that ''b>0''
|-
| <math> \log_b(b) = 1 \!\, </math> || because || <math> b^1 = b\!\, </math>
|}
 
Note that log<sub>''b''</sub>(0) is undefined because there is no number ''x'' such that ''b''<sup>''x''</sup>&nbsp;=&nbsp;0. In fact, there is a [[vertical asymptote]] on the graph of log<sub>''b''</sub>(''x'') at ''x''&nbsp;=&nbsp;0.
 
=== Canceling exponentials ===
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).
 
: <math> b^{\log_b(x)} = x\text{ because }\operatorname{antilog}_b(\log_b(x)) = x \, </math>
 
: <math> \log_b(b^x) = x\text{ because }\log_b(\operatorname{antilog}_b(x)) = x \, </math>
 
Both of the above are derived from the following two equations that define a logarithm:-
 
: <math> b^c = x\text{, }\log_b(x) = c \, </math>
 
Substituting c in the left equation gives b<big><sup>log<sub>b</sub>(x)</sup></big>&nbsp;=&nbsp;x, and substituting x in the right gives log<sub>b</sub>(b<big><sup>c</sup></big>)&nbsp;=&nbsp;c. Finally, replace c by x.
 
=== Using simpler operations ===
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}}.
 
{| cellpadding=3
| <math> \log_b(xy) = \log_b(x) + \log_b(y) \!\, </math> || because || <math> b^c \cdot b^d = b^{c + d} \!\, </math>
|-
| <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>
|-
| <math> \log_b(x^d) = d \log_b(x) \!\, </math> || because || <math> (b^c)^d = b^{cd} \!\, </math>
|-
| <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>
|-
| <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>
|-
| <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>
 
|}
 
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.
 
The laws result from canceling exponentials and appropriate law of indices. Starting with the first law:
 
<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>
 
The law for powers exploits another of the laws of indices:
 
<math>x^y = (b^{\log_b(x)})^y = b^{y \log_b(x)} \Rightarrow \log_b(x^y) = y \log_b(x)</math>
 
The law relating to quotients then follows:
 
<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>
 
Similarly, the root law is derived by rewriting the root as a reciprocal power:
 
<math>\log_b(\sqrt[y]x) = \log_b(x^{\frac{1}{y}}) = \frac{1}{y}\log_b(x)</math>
 
=== Changing the base ===
:<math>\log_b a = {\log_d a \over \log_d b}</math>
 
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).
 
==== Proof ====
 
:Let <math>c=\log_b a</math>.
 
:Then <math>b^c=a</math>.
 
:Take <math>\log_d</math> on both sides: <math>\log_d b^c=\log_d a</math>
 
:Simplify and solve for <math>c</math>: <math> c\log_d b=\log_d a</math>
 
:<math>c=\frac{\log_d a}{\log_d b}</math>
 
:Since <math>c=\log_b a</math>, then <math>\log_b a=\frac{\log_d a}{\log_d b}</math>
 
This formula has several consequences:
 
:<math> \log_b a = \frac {1} {\log_a b} </math>
 
:<math> \log_{b^n} a =  {{\log_b a} \over n} </math>
 
:<math> b^{\log_a d} = d^{\log_a b} </math>
 
:<math>- \log_b a = \log_b \left({1 \over a}\right) = \log_{1 \over b} a</math>
 
<!-- extra blank space between two lines of "displayed" [[TeX]] for legibility -->
 
:<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>
 
where <math>\scriptstyle\pi\,</math> is any [[permutation]] of the subscripts 1,&nbsp;...,&nbsp;''n''.  For example
 
:<math> \log_b w\cdot \log_a x\cdot \log_d c\cdot \log_d z
= \log_d w\cdot \log_b x\cdot \log_a c\cdot \log_d z. \, </math>
 
=== Summation/subtraction ===
The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:
:<math>\log_b (a+c) = \log_b a + \log_b (1+b^{\log_b c - \log_b a})</math>
 
:<math>\log_b (a-c) = \log_b a + \log_b (1-b^{\log_b c - \log_b a})</math>
 
which gives the special cases:
 
:<math>\log_b (a+c) = \log_b a + \log_b \left(1+\frac{c}{a}\right)</math>
 
:<math>\log_b (a-c) = \log_b a + \log_b \left(1-\frac{c}{a}\right)</math>
 
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.
 
More generally:
:<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>
 
where <math>a_0,\ldots ,a_N > 0</math>.
 
=== Exponents ===
A useful identity involving exponents:
:<math> x^{\frac{\log(\log(x))}{\log(x)}} = \log(x) </math>
 
== Calculus identities ==
=== [[Limit of a function|Limits]] ===
:<math>\lim_{x \to 0^+} \log_a x = -\infty \quad \mbox{if } a > 1</math>
 
:<math>\lim_{x \to 0^+} \log_a x =  +\infty \quad \mbox{if } a < 1</math>
 
:<math>\lim_{x \to+\infty} \log_a x=  +\infty \quad \mbox{if } a > 1</math>
 
:<math>\lim_{x \to+\infty} \log_a x=  -\infty \quad \mbox{if } a < 1</math>
 
:<math>\lim_{x \to 0^+} x^b \log_a x = 0 \quad \mbox{if } b > 0</math>
 
:<math>\lim_{x \to+\infty} {1 \over x^b} \log_a x = 0 \quad \mbox{if } b > 0</math>
 
The last limit is often summarized as "logarithms grow more slowly than any power or root of ''x''".
 
=== [[Derivative]]s of logarithmic functions ===
:<math>{d \over dx} \ln x = {1 \over x },</math>
:<math>{d \over dx} \log_b x = {1 \over x \ln b},</math>
Where <math>x > 0</math>, <math>b > 0</math>, and <math>b \ne 1</math>.
 
=== Integral definition ===
:<math>\ln x = \int_1^x \frac {1}{t} dt </math>
 
=== [[Integral]]s of logarithmic functions ===
: <math>\int \log_a x \, dx = x(\log_a x - \log_a e) + C</math>
 
To remember higher integrals, it's convenient to define:
:<math>x^{\left [n \right]} = x^{n}(\log(x) - H_n)</math>
Where <math>H_n</math> is the nth [[Harmonic number]].
 
:<math>x^{\left [ 0 \right ]} = \log x</math>
:<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>
:<math>x^{\left [ 3 \right ]} = x^3 \log(x) - \begin{matrix} \frac{11}{6} \end{matrix} \, x^3</math>
 
Then,
:<math>\frac {d}{dx} \, x^{\left [ n \right ]} = n \, x^{\left [ n-1 \right ]}</math>
:<math>\int x^{\left [ n \right ]}\,dx = \frac {x^{\left [ n+1 \right ]}} {n+1} + C</math>
 
== Approximating large numbers ==
 
The identities of logarithms can be used to approximate large numbers. Note that log<sub>''b''</sub>(''a'')&nbsp;+&nbsp;log<sub>''b''</sub>(''c'') =&nbsp;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>&nbsp;&minus;&nbsp;1. To get the base-10 logarithm, we would multiply 32,582,657 by log<sub>10</sub>(2), getting 9,808,357.09543 =&nbsp;9,808,357&nbsp;+&nbsp;0.09543. We can then get 10<sup>9,808,357</sup>&nbsp;&times;&nbsp;10<sup>0.09543</sup> ≈&nbsp;1.25&nbsp;&times;&nbsp;10<sup>9,808,357</sup>.
 
Similarly, factorials can be approximated by summing the logarithms of the terms.
 
== Complex logarithm identities ==
 
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]].
 
=== Definitions ===
 
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.
 
:ln(''r'') is the standard natural logarithm of the real number ''r''.
:Log(''z'') is the principal value of the complex logarithm function and has imaginary part in the range (-π, π].
: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'').
 
:<math>\operatorname{Log}(z) = \ln(|z|) + i \operatorname{Arg}(z)</math>
:<math>e^{\operatorname{Log}(z)} = z</math>
 
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.
 
:log(''z'') is the set of complex numbers ''v'' which satisfy e<sup>''v''</sup> = ''z''
:arg(''z'') is the set of possible values of the [[Arg (mathematics)|arg]] function applied to ''z''.
 
When ''k'' is any integer:
 
:<math>\log(z) = \ln(|z|) + i \arg(z)</math>
:<math>\log(z) = \operatorname{Log}(z) + 2 \pi i k</math>
:<math>e^{\log(z)} = z</math>
 
=== Constants ===
 
Principal value forms:
 
:<math>\operatorname{Log}(1) = 0</math>
:<math>\operatorname{Log}(e) = 1</math>
 
Multiple value forms, for any ''k'' an integer:
 
:<math>\log(1) = 0 + 2 \pi i k</math>
:<math>\log(e) = 1 + 2 \pi i k</math>
 
=== Summation ===
 
Principal value forms:
 
:<math>\operatorname{Log}(z_1) + \operatorname{Log}(z_2) = \operatorname{Log}(z_1 z_2) \pmod {2 \pi i}</math>
:<math>\operatorname{Log}(z_1) - \operatorname{Log}(z_2) = \operatorname{Log}(z_1 / z_2) \pmod {2 \pi i}</math>
 
Multiple value forms:
 
:<math>\log(z_1) + \log(z_2) = \log(z_1 z_2)</math>
:<math>\log(z_1) - \log(z_2) = \log(z_1 / z_2)</math>
 
=== Powers ===
 
A complex power of a complex number can have many possible values.
 
Principal value form:
 
:<math>{z_1}^{z_2} = e^{z_2 \operatorname{Log}(z_1)} </math>
 
:<math>\operatorname{Log}{\left({z_1}^{z_2}\right)} = z_2 \operatorname{Log}(z_1) \pmod {2 \pi i}</math>
 
Multiple value forms:
 
:<math>{z_1}^{z_2} = e^{z_2 \log(z_1)}</math>
 
Where ''k''<sub>1</sub>, ''k''<sub>2</sub> are any integers:
 
:<math>\log{\left({z_1}^{z_2}\right)} = z_2 \log(z_1) + 2 \pi i k_2</math>
:<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>
 
== See also ==
* [[List of trigonometric identities]]
* [[Exponential function]]
 
== References ==
{{reflist}}
 
== External links ==
* {{MathWorld|Logarithm|Logarithm}}
* [http://www.mathwords.com/l/logarithm.htm Logarithm] in Mathwords
 
[[Category:Logarithms]]
[[Category:Mathematical identities]]
[[Category:Articles containing proofs]]

Revision as of 20:00, 4 March 2014


mindtools.comAfter I let the kava tea steep and brew for about 10 minutes to make sure I get all the calming and relaxing chemicals out of the tea bag, and than I simply sip on the tea until my server anxiety attacks go away. I strongly suggest you click the links on this page to learn exactly how the Panic Away anxiety treatment program can benefit you and your life. The biggest benefit of the Panic Away program is that there is no need to visit a doctor, and there are no costly therapist bills to pay. This could not be further from the truth... Panic attacks are one of the easiest disorders to treat in the psychiatric world. This website continues to be online in excess of seven years and contains not received even one complaint from the clients. What to eat and what to avoid. These numerous arousals caused by sleep apnea, can sometimes trigger anxiety attacks.
He came up with this approach named the 'One Move' Technique. The way I make it is simply by boiling some water on top of my stove and than by simply pouring the boiling water over a kava tea bag in my tea cup. Panic is an innate flight or fight response to danger or intimidation that one is experiencing. However, this is not a final solution for this problem, rather it is just a way of hiding it. Another change in my diet was giving up the junk food and trying to eat healthier. The very first panic attack experience one has can pass negative effects on to a one's psyche; most people do not fully recover from such an episode. However, it is essential to consult your physician before taking the medicines.
What inspired that costume and mask? When your body is out of balance it will send distress signals to your mind. The first is to get rid of your panic attacks. Over the last couple years of my life I have been developing more and more panic and anxiety attacks due to the amount of stress in my every day life, and since I am into alternative medicines I like to use natural herbal remedies to help relieve my stress from my every day life so I do not have panic or anxiety attacks. When you do something like this it distracts you from your original thoughts and anxieties. Another thing that is helpful for panic attacks is to eliminate harmful substances from your life. Just keep reminding yourself that the feeling will pass. Panic Away System - How does it Work?
many people believe that they will help take the edge off which in turn will help stop panic attacks. This may not only affect their social life, but also have a negative impact on their relationships. Natural treatments help in this case, but they will not fix the problem for good. However the fact is that this product seems to be helping a lot of people. Comments and Testimonials Comments and testimonials are available here. At this time in my life I was only comfortable traveling around a 5 mile radius from my home, which I would come to find out through my education that this is quite common for people with my disorder. Take psychotherapy sessions from a psychiatrist and try to find out the reasons as to why you're experiencing anxiety and panic attacks. This was a time during the day I looked forward to. Once you master it, then your response to panic attacks will be natural and seamless.
When you are exhausted, one's body is extended to its complete control. Before going into the causes and its impact in details, let us see a summary of this economic recession so that you have an idea as to what it actually was. Try to learn how to surf through the anxiety and the panic attacks. This individual then fears that precise place or event. All natural anti anxiety medicines include some herbs to retain positive mood balance, promote relaxation and support healthy neurotransmitter balance. panic away book. Prescribed Natural Medication for Panic Treatment Anxiety is an emotional imbalance that affects both the mind and body. With no real attachments, Doug never has to fear losing anyone close to him. If only it was that easy to recognize a tsunami coming, then many would prevent the damage.

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