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In mathematics, there are many ..

4Changing the base
6Calculus identities
8Complex logarithm identities

Trivial identities[edit]

${displaystyle log _{b}(1)=0}$	because	${displaystyle b^{0}=1}$ , given that b doesn't equal 0
${displaystyle log _{b}(b)=1}$	because	${displaystyle b^{1}=b}$

Cancelling exponentials[edit]

Logarithms and exponentials 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).

{displaystyle b^{log _{b}(x)}=x{text{ because }}{mbox{antilog}}_{b}(log _{b}(x))=x}

{displaystyle log _{b}(b^{x})=x{text{ because }}log _{b}({mbox{antilog}}_{b}(x))=x}

Both of the above are derived from the following two equations that define a logarithm:-

{displaystyle b^{c}=x{text{, }}log _{b}(x)=c}

Substituting c in the left equation gives b^log_b(x) = x, and substituting x in the right gives log_b(b^c) = c. Finally, replace c by x.

Using simpler operations[edit]

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 x = b^c, and/or y = b^d so that log_b(x) = c and log_b(y) = d. Derivations also use the log definitions x = b^log_b(x) and x = log_b(b^x).

${displaystyle log _{b}(xy)=log _{b}(x)+log _{b}(y)}$	because	${displaystyle b^{c}cdot b^{d}=b^{c+d}}$
${displaystyle log _{b}({tfrac {x}{y}})=log _{b}(x)-log _{b}(y)}$	because	${displaystyle {tfrac {b^{c}}{b^{d}}}=b^{c-d}}$
${displaystyle log _{b}(x^{d})=dlog _{b}(x)}$	because	${displaystyle (b^{c})^{d}=b^{cd}}$
${displaystyle log _{b}left({sqrt[{y}]{x}}right)={frac {log _{b}(x)}{y}}}$	because	${displaystyle {sqrt[{y}]{x}}=x^{1/y}}$
${displaystyle x^{log _{b}(y)}=y^{log _{b}(x)}}$	because	${displaystyle x^{log _{b}(y)}=b^{log _{b}(x)log _{b}(y)}=(b^{log _{b}(y)})^{log _{b}(x)}=y^{log _{b}(x)}}$
${displaystyle clog _{b}(x)+dlog _{b}(y)=log _{b}(x^{c}y^{d})}$	because	${displaystyle log _{b}(x^{c}y^{d})=log _{b}(x^{c})+log _{b}(y^{d})}$

Where ${displaystyle b}$ , ${displaystyle x}$ , and ${displaystyle y}$ are positive real numbers and ${displaystyle bneq 1}$ . Both ${displaystyle c}$ and ${displaystyle d}$ are real numbers.

The laws result from canceling exponentials and appropriate law of indices. Starting with the first law:

${displaystyle xy=b^{log _{b}(x)}b^{log _{b}(y)}=b^{log _{b}(x)+log _{b}(y)}Rightarrow log _{b}(xy)=log _{b}(b^{log _{b}(x)+log _{b}(y)})=log _{b}(x)+log _{b}(y)}$

The law for powers exploits another of the laws of indices:

${displaystyle x^{y}=(b^{log _{b}(x)})^{y}=b^{ylog _{b}(x)}Rightarrow log _{b}(x^{y})=ylog _{b}(x)}$

The law relating to quotients then follows:

${displaystyle log _{b}{bigg (}{frac {x}{y}}{bigg )}=log _{b}(xy^{-1})=log _{b}(x)+log _{b}(y^{-1})=log _{b}(x)-log _{b}(y)}$

${displaystyle log _{b}{bigg (}{frac {1}{y}}{bigg )}=log _{b}(y^{-1})=-log _{b}(y)}$

Similarly, the root law is derived by rewriting the root as a reciprocal power:

${displaystyle log _{b}({sqrt[{y}]{x}})=log _{b}(x^{frac {1}{y}})={frac {1}{y}}log _{b}(x)}$

Changing the base[edit]

{displaystyle log _{b}a={frac {log _{d}(a)}{log _{d}(b)}}}

This identity is useful to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log₁₀, but not all calculators have buttons for the logarithm of an arbitrary base.

Consider the equation

{displaystyle b^{c}=a}

Take logarithm base

{displaystyle d}

of both sides:

{displaystyle log _{d}b^{c}=log _{d}a}

Simplify and solve for

{displaystyle c}

{displaystyle clog _{d}b=log _{d}a}

{displaystyle c={frac {log _{d}a}{log _{d}b}}}

Since

{displaystyle c=log _{b}a}

, then

{displaystyle log _{b}a={frac {log _{d}a}{log _{d}b}}}

This formula has several consequences:

{displaystyle log _{b}a={frac {1}{log _{a}b}}}

{displaystyle log _{b^{n}}a={{log _{b}a} over n}}

{displaystyle b^{log _{a}d}=d^{log _{a}b}}

- \log_{b} a = \log_{b} (\frac{1}{a}) = \log_{\frac{1}{b}} a {displaystyle -log _{b}a=log _{b}left({1 over a}right)=log _{1 over b}a}

{displaystyle log _{b_{1}}a_{1},cdots ,log _{b_{n}}a_{n}=log _{b_{pi (1)}}a_{1},cdots ,log _{b_{pi (n)}}a_{n},}

where ${displaystyle scriptstyle pi }$ is any permutation of the subscripts 1, ..., n. For example

{displaystyle log _{b}wcdot log _{a}xcdot log _{d}ccdot log _{d}z=log _{d}wcdot log _{b}xcdot log _{a}ccdot log _{d}z.}

Summation/subtraction[edit]

The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:

{displaystyle log _{b}(a+c)=log _{b}a+log _{b}left({frac {c}{a}}+1right)}

{displaystyle log _{b}(a-c)=log _{b}a+log _{b}left({frac {c}{a}}-1right)}

Note that in practice ${displaystyle a}$ and ${displaystyle c}$ have to be switched on the right hand side of the equations if ${displaystyle c>a}$ . Also note that the subtraction identity is not defined if ${displaystyle a=c}$ since the logarithm of zero is not defined. Many programming languages have a specific log1p(x) function that calculates ${displaystyle log _{e}(1+x)}$ without underflow when ${displaystyle x}$ is small.

More generally:

{displaystyle 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)}

where ${displaystyle a_{0}>a_{1}>ldots >a_{N}}$ are sorted in descending order.

Exponents[edit]

A useful identity involving exponents:

{displaystyle x^{frac {log(log(x))}{log(x)}}=log(x)}

or more universally:

{displaystyle x^{frac {log(a)}{log(x)}}=a}

Other/Resulting Identities[edit]

{displaystyle {frac {1}{{frac {1}{log _{x}(a)}}+{frac {1}{log _{y}(a)}}}}=log _{xy}(a)}

Inequalities[edit]

Based on ^[1] , ^[2] and ^[3]

{displaystyle {frac {x}{1+x}}leq ln(1+x)leq {frac {x(6+x)}{6+4x}}leq x{mbox{ for all }}-1<x}

{displaystyle {begin{aligned}{frac {2x}{2+x}}&leq 3-{sqrt {frac {27}{3+2x}}}leq {frac {x}{sqrt {1+x+x^{2}/12}}}&leq ln(1+x)leq {frac {x}{sqrt {1+x}}}leq {frac {x}{2}}{frac {2+x}{1+x}}&{mbox{ for }}0leq x{mbox{, reverse for }}-1<xleq 0end{aligned}}}

All are accurate around ${displaystyle x=0}$ , but not for large numbers.

Calculus identities[edit]

Limits[edit]

{displaystyle lim _{xto 0^{+}}log _{a}(x)=-infty quad {mbox{if }}a>1}

{displaystyle lim _{xto 0^{+}}log _{a}(x)=infty quad {mbox{if }}0<semantics><mrow><mstyle displaystyle=

{displaystyle lim _{xto infty }log _{a}(x)=infty quad {mbox{if }}a>1}

{displaystyle lim _{xto infty }log _{a}(x)=-infty quad {mbox{if }}0<semantics><mrow><mstyle displaystyle=

{displaystyle lim _{xto 0^{+}}x^{b}log _{a}(x)=0quad {mbox{if }}b>0}

{displaystyle lim _{xto infty }{frac {log _{a}(x)}{x^{b}}}=0quad {mbox{if }}b>0}

The last limit is often summarized as 'logarithms grow more slowly than any power or root of x'.

Derivatives of logarithmic functions[edit]

\frac{d}{d x} \ln x = \frac{1}{x}, {displaystyle {d over dx}ln x={1 over x},}

\frac{d}{d x} \log_{b} x = \frac{1}{x \ln b}, {displaystyle {d over dx}log _{b}x={1 over xln b},}

Where ${displaystyle x>0}$ , ${displaystyle b>0}$ , and ${displaystyle bneq 1}$ .

Integral definition[edit]

{displaystyle ln x=int _{1}^{x}{frac {1}{t}}dt}

Integrals of logarithmic functions[edit]

{displaystyle int log _{a}x,dx=x(log _{a}x-log _{a}e)+C}

To remember higher integrals, it's convenient to define:

{displaystyle x^{left[nright]}=x^{n}(log(x)-H_{n})}

Where ${displaystyle H_{n}}$ is the nth Harmonic number.

{displaystyle x^{left[0right]}=log x}

{displaystyle x^{left[1right]}=xlog(x)-x}

x^{[2]} = x^{2} \log (x) - \begin{matrix} \frac{3}{2} \end{matrix} x^{2} {displaystyle x^{left[2right]}=x^{2}log(x)-{begin{matrix}{frac {3}{2}}end{matrix}},x^{2}}

x^{[3]} = x^{3} \log (x) - \begin{matrix} \frac{11}{6} \end{matrix} x^{3} {displaystyle x^{left[3right]}=x^{3}log(x)-{begin{matrix}{frac {11}{6}}end{matrix}},x^{3}}

Then,

{displaystyle {frac {d}{dx}},x^{left[nright]}=n,x^{left[n-1right]}}

{displaystyle int x^{left[nright]},dx={frac {x^{left[n+1right]}}{n+1}}+C}

Approximating large numbers[edit]

The identities of logarithms can be used to approximate large numbers. Note that log_b(a) + log_b(c) = log_b(ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2^32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log₁₀(2), getting 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 10^9,808,357 × 10^0.09543 ≈ 1.25 × 10^9,808,357.

Similarly, factorials can be approximated by summing the logarithms of the terms.

Complex logarithm identities[edit]

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[edit]

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 function, its value is restricted to (-π, π]. It can be computed using Arg(x+iy)= atan2(y, x).

{displaystyle operatorname {Log} (z)=ln( z )+ioperatorname {Arg} (z)}

{displaystyle e^{operatorname {Log} (z)}=z}

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^v = z

arg(z) is the set of possible values of the arg function applied to z.

When k is any integer:

{displaystyle log(z)=ln( z )+iarg(z)}

{displaystyle log(z)=operatorname {Log} (z)+2pi ik}

{displaystyle e^{log(z)}=z}

Constants[edit]

Principal value forms:

{displaystyle operatorname {Ln} (1)=0}

{displaystyle operatorname {Ln} (e)=1}

Multiple value forms, for any k an integer:

{displaystyle log(1)=0+2pi ik}

{displaystyle log(e)=1+2pi ik}

Summation[edit]

Principal value forms:

{displaystyle operatorname {Log} (z_{1})+operatorname {Log} (z_{2})=operatorname {Log} (z_{1}z_{2}){pmod {2pi i}}}

{displaystyle operatorname {Log} (z_{1})-operatorname {Log} (z_{2})=operatorname {Log} (z_{1}/z_{2}){pmod {2pi i}}}

Multiple value forms:

{displaystyle log(z_{1})+log(z_{2})=log(z_{1}z_{2})}

{displaystyle log(z_{1})-log(z_{2})=log(z_{1}/z_{2})}

Powers[edit]

A complex power of a complex number can have many possible values.

Principal value form:

{displaystyle {z_{1}}^{z_{2}}=e^{z_{2}operatorname {Log} (z_{1})}}

{displaystyle operatorname {Log} {left({z_{1}}^{z_{2}}right)}=z_{2}operatorname {Log} (z_{1}){pmod {2pi i}}}

Multiple value forms:

{displaystyle {z_{1}}^{z_{2}}=e^{z_{2}log(z_{1})}}

Where k₁, k₂ are any integers:

{displaystyle log {left({z_{1}}^{z_{2}}right)}=z_{2}log(z_{1})+2pi ik_{2}}

{displaystyle log {left({z_{1}}^{z_{2}}right)}=z_{2}operatorname {Log} (z_{1})+z_{2}2pi ik_{1}+2pi ik_{2}}

References[edit]

^http://ajmaa.org/RGMIA/papers/v7n2/pade.pdf
^http://www.lkozma.net/inequalities_cheat_sheet/ineq.pdf
^http://downloads.hindawi.com/archive/2013/412958.pdf

External links[edit]

Weisstein, Eric W.'Logarithm'. MathWorld.
Logarithm in Mathwords

Retrieved from 'https://en.wikipedia.org/w/index.php?title=List_of_logarithmic_identities&oldid=910901725'

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B Of A Login

Trivial identities[edit]

Cancelling exponentials[edit]

Using simpler operations[edit]

Changing the base[edit]

Summation/subtraction[edit]

Exponents[edit]

Other/Resulting Identities[edit]

Inequalities[edit]

Calculus identities[edit]

Limits[edit]

Derivatives of logarithmic functions[edit]

Integral definition[edit]

Integrals of logarithmic functions[edit]

Approximating large numbers[edit]

Complex logarithm identities[edit]

Definitions[edit]

Constants[edit]

Summation[edit]

Powers[edit]

See also[edit]

References[edit]

External links[edit]

Recent Posts

B Of A Login

Trivial identities[edit]

Cancelling exponentials[edit]

Using simpler operations[edit]

Changing the base[edit]

Summation/subtraction[edit]

Exponents[edit]

Other/Resulting Identities[edit]

Inequalities[edit]

Calculus identities[edit]

Limits[edit]

Derivatives of logarithmic functions[edit]

Integral definition[edit]

Integrals of logarithmic functions[edit]

Approximating large numbers[edit]

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Complex logarithm identities[edit]

Definitions[edit]

Constants[edit]

Summation[edit]

Powers[edit]

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See also[edit]

References[edit]

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External links[edit]

Recent Posts