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What Is Log Z Complex Formula?

Published Aug 29, 2025 9 min read
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The complex logarithm formula for a non-zero complex number z=x+iyz equals x plus i y 𝑧=π‘₯+𝑖𝑦 is given by:log(z)=ln|z|+iarg(z)=lnx2+y2+i(ΞΈ+2Ο€n)log z equals l n the absolute value of z end-absolute-value plus i arg z equals l n the square root of x squared plus y squared end-root plus i open paren theta plus 2 pi n close paren

log(𝑧)=ln|𝑧|+𝑖arg(𝑧)=lnπ‘₯2+𝑦2√+𝑖(πœƒ+2πœ‹π‘›)

where:

  • ln|z|l n the absolute value of z end-absolute-value

    ln|𝑧|

    is the standard real-valued natural logarithm of the modulus (or magnitude) of zz

    𝑧

    .

  • arg(z)arg z

    arg(𝑧)

    is the multi-valued argument (or angle) of zz

    𝑧

    .

  • ΞΈtheta

    πœƒ

    is the principal value of the argument, typically restricted to the interval βˆ’Ο€<θ≀πnegative pi is less than theta is less than or equal to pi

    βˆ’πœ‹<πœƒβ‰€πœ‹

    .

  • nn

    𝑛

    is any integer (n=0,Β±1,Β±2,…n equals 0 comma plus or minus 1 comma plus or minus 2 comma …

    𝑛=0,Β±1,Β±2,…

    ).

This multi-valued nature of the complex logarithm is a direct consequence of the periodic nature of the complex exponential function.

Why the logarithm is multi-valued

To understand why the complex logarithm has infinite possible values, consider the inverse relationship between the logarithm and the exponential function. For a complex number w=u+ivw equals u plus i v

𝑀=𝑒+𝑖𝑣

to be a logarithm of a non-zero complex number z=x+iyz equals x plus i y

𝑧=π‘₯+𝑖𝑦

, it must satisfy ew=ze to the w-th power equals z

𝑒𝑀=𝑧

.

Using the properties of the complex exponential, we can write:ew=eu+iv=eueiv=eu(cosv+isinv)e to the w-th power equals e raised to the u plus i v power equals e to the u-th power e raised to the i v power equals e to the u-th power open paren cosine v plus i sine v close paren

𝑒𝑀=𝑒𝑒+𝑖𝑣=𝑒𝑒𝑒𝑖𝑣=𝑒𝑒(cos𝑣+𝑖sin𝑣)

We also know that the complex number zz

𝑧

can be written in polar form as z=|z|(cosΞΈ+isinΞΈ)z equals the absolute value of z end-absolute-value open paren cosine theta plus i sine theta close paren

𝑧=|𝑧|(cosπœƒ+𝑖sinπœƒ)

, where |z|=x2+y2the absolute value of z end-absolute-value equals the square root of x squared plus y squared end-root

|𝑧|=π‘₯2+𝑦2√

and ΞΈ=arg(z)theta equals arg z

πœƒ=arg(𝑧)

.

By equating the two expressions for zz

𝑧

, we get:eu(cosv+isinv)=|z|(cosΞΈ+isinΞΈ)e to the u-th power open paren cosine v plus i sine v close paren equals the absolute value of z end-absolute-value open paren cosine theta plus i sine theta close paren

𝑒𝑒(cos𝑣+𝑖sin𝑣)=|𝑧|(cosπœƒ+𝑖sinπœƒ)

This implies that the magnitudes must be equal, and the arguments must be equivalent up to an integer multiple of 2Ο€2 pi

2πœ‹

:eu=|z|⟹u=ln|z|e to the u-th power equals the absolute value of z end-absolute-value ⟹ u equals l n the absolute value of z end-absolute-value

𝑒𝑒=|𝑧|βŸΉπ‘’=ln|𝑧|

v=ΞΈ+2Ο€nfor any integer nv equals theta plus 2 pi n space for any integer n

𝑣=πœƒ+2πœ‹π‘›foranyinteger𝑛

This gives the multi-valued formula for the complex logarithm, w=u+ivw equals u plus i v

𝑀=𝑒+𝑖𝑣

:log(z)=ln|z|+i(ΞΈ+2Ο€n)log z equals l n the absolute value of z end-absolute-value plus i open paren theta plus 2 pi n close paren

log(𝑧)=ln|𝑧|+𝑖(πœƒ+2πœ‹π‘›)

The principal value of the logarithm

To deal with the multi-valued nature of the complex logarithm, a single value is often selected, called the principal value, denoted by Log(z)cap L o g space open paren z close paren

Log(𝑧)

. This is achieved by restricting the argument to a specific interval. The standard choice for the principal value is to restrict the argument Arg(z)cap A r g space open paren z close paren

Arg(𝑧)

to the range (βˆ’Ο€,Ο€]open paren negative pi comma pi close bracket

(βˆ’πœ‹,πœ‹]

.

The formula for the principal value of the complex logarithm is:Log(z)=ln|z|+iArg(z)where βˆ’Ο€<Arg(z)≀πcap L o g space open paren z close paren equals l n the absolute value of z end-absolute-value plus i space cap A r g space open paren z close paren space where minus pi is less than space cap A r g space open paren z close paren is less than or equal to pi

Log(𝑧)=ln|𝑧|+𝑖Arg(𝑧)whereβˆ’πœ‹<Arg(𝑧)β‰€πœ‹

For example, for z=iz equals i

𝑧=𝑖

, the polar form is i=1β‹…eiΟ€/2i equals 1 center dot e raised to the i pi / 2 power

𝑖=1β‹…π‘’π‘–πœ‹/2

. The modulus is |i|=1the absolute value of i end-absolute-value equals 1

|𝑖|=1

and the principal argument is Arg(i)=Ο€/2cap A r g space open paren i close paren equals pi / 2

Arg(𝑖)=πœ‹/2

. Thus:Log(i)=ln(1)+i(Ο€/2)=iΟ€2cap L o g space open paren i close paren equals l n 1 plus i open paren pi / 2 close paren equals i the fraction with numerator pi and denominator 2 end-fraction

Log(𝑖)=ln(1)+𝑖(πœ‹/2)=π‘–πœ‹2

The set of all possible values for log(i)log i

log(𝑖)

is i(Ο€/2+2Ο€n)i open paren pi / 2 plus 2 pi n close paren

𝑖(πœ‹/2+2πœ‹π‘›)

for integer nn

𝑛

.

Branch cuts and branch points

The complex logarithm function is not continuous on the entire complex plane. For instance, the principal logarithm Log(z)cap L o g space open paren z close paren

Log(𝑧)

is discontinuous along the negative real axis. This is because as a point approaches the negative real axis from above, its argument approaches Ο€pi

πœ‹

. As it approaches from below, its argument approaches βˆ’Ο€negative pi

βˆ’πœ‹

. This abrupt jump of 2Ο€2 pi

2πœ‹

in the imaginary part causes a discontinuity.

This line of discontinuity is called a branch cut. The point at the origin, z=0z equals 0

𝑧=0

, is called a branch point because any closed path around the origin will cause the value of the logarithm to change by a multiple of 2Ο€i2 pi i

2πœ‹π‘–

. Different branches of the logarithm can be defined by choosing different intervals for the argument, leading to different branch cuts.

Properties of the complex logarithm

The complex logarithm shares many properties with its real counterpart, but care must be taken due to its multi-valued nature.

  • Inverse Relationship: The fundamental property elog(z)=ze raised to the log z power equals z

    𝑒log(𝑧)=𝑧

    holds true for any branch of the logarithm. However, log(ez)=z+2Ο€inlog open paren e to the z-th power close paren equals z plus 2 pi i n

    log(𝑒𝑧)=𝑧+2πœ‹π‘–π‘›

    for some integer nn

    𝑛

    , not necessarily just zz

    𝑧

    .

  • Multiplication: The rule log(z1z2)=log(z1)+log(z2)log open paren z sub 1 z sub 2 close paren equals log open paren z sub 1 close paren plus log open paren z sub 2 close paren

    log(𝑧1𝑧2)=log(𝑧1)+log(𝑧2)

    holds only up to an integer multiple of 2Ο€i2 pi i

    2πœ‹π‘–

    . This means that the principal value identity Log(z1z2)=Log(z1)+Log(z2)cap L o g space open paren z sub 1 z sub 2 close paren equals space cap L o g space open paren z sub 1 close paren plus space cap L o g space open paren z sub 2 close paren

    Log(𝑧1𝑧2)=Log(𝑧1)+Log(𝑧2)

    is not always true. For example, let z1=-1z sub 1 equals negative 1

    𝑧1=βˆ’1

    and z2=-1z sub 2 equals negative 1

    𝑧2=βˆ’1

    .

    • Log(z1)=Log(-1)=ln(1)+iΟ€=iΟ€cap L o g space open paren z sub 1 close paren equals space cap L o g space open paren negative 1 close paren equals l n 1 plus i pi equals i pi

      Log(𝑧1)=Log(βˆ’1)=ln(1)+π‘–πœ‹=π‘–πœ‹

      .

    • Log(z2)=Log(-1)=ln(1)+iΟ€=iΟ€cap L o g space open paren z sub 2 close paren equals space cap L o g space open paren negative 1 close paren equals l n 1 plus i pi equals i pi

      Log(𝑧2)=Log(βˆ’1)=ln(1)+π‘–πœ‹=π‘–πœ‹

      .

    • Log(z1z2)=Log(1)=ln(1)+i(0)=0cap L o g space open paren z sub 1 z sub 2 close paren equals space cap L o g space open paren 1 close paren equals l n 1 plus i open paren 0 close paren equals 0

      Log(𝑧1𝑧2)=Log(1)=ln(1)+𝑖(0)=0

      .

    • Here, Log(-1)+Log(-1)=iΟ€+iΟ€=2iΟ€cap L o g space open paren negative 1 close paren plus space cap L o g space open paren negative 1 close paren equals i pi plus i pi equals 2 i pi

      Log(βˆ’1)+Log(βˆ’1)=π‘–πœ‹+π‘–πœ‹=2π‘–πœ‹

      , which is not equal to Log(1)=0cap L o g space open paren 1 close paren equals 0

      Log(1)=0

      . The difference is 2Ο€i2 pi i

      2πœ‹π‘–

      .

  • Division: Similarly, log(z1/z2)=log(z1)βˆ’log(z2)log open paren z sub 1 / z sub 2 close paren equals log open paren z sub 1 close paren minus log open paren z sub 2 close paren

    log(𝑧1/𝑧2)=log(𝑧1)βˆ’log(𝑧2)

    also holds up to an integer multiple of 2Ο€i2 pi i

    2πœ‹π‘–

    .

  • Logarithm of a Complex Power: For zβ‰ 0z is not equal to 0

    𝑧≠0

    and a∈Ca is an element of the complex numbers

    π‘Žβˆˆβ„‚

    , the complex power zaz to the a-th power

    π‘§π‘Ž

    is defined as ealog(z)e raised to the a log z power

    π‘’π‘Žlog(𝑧)

    . Because of the multi-valued nature of log(z)log z

    log(𝑧)

    , zaz to the a-th power

    π‘§π‘Ž

    can also be multi-valued.

Applications

The complex logarithm is an essential concept in complex analysis with numerous applications:

  • Complex Exponentiation: It is used to define complex powers, such as aba to the b-th power

    π‘Žπ‘

    where both aa

    π‘Ž

    and bb

    𝑏

    are complex numbers.

  • Solving Equations: It allows for the solution of equations involving complex exponentials.

  • Mapping: The principal logarithm function Log(z)cap L o g space open paren z close paren

    Log(𝑧)

    maps the punctured complex plane (excluding the origin) to a horizontal strip in the complex plane. It's a conformal map, preserving angles between intersecting curves.

  • Inverse Laplace Transform: It is an essential tool in computing inverse Laplace transforms in certain contexts.

  • Inverse Trigonometric and Hyperbolic Functions: These functions can be expressed in terms of complex logarithms. For example:

    • arcsin(z)=βˆ’iLog(iz+1βˆ’z2)arc sine z equals negative i space cap L o g space open paren i z plus the square root of 1 minus z squared end-root close paren

      arcsin(𝑧)=βˆ’π‘–Log(𝑖𝑧+1βˆ’π‘§2√)

    • arctan(z)=12iLog(1+iz1βˆ’iz)arc tangent z equals 1 over 2 i end-fraction space cap L o g space open paren the fraction with numerator 1 plus i z and denominator 1 minus i z end-fraction close paren

      arctan(𝑧)=12𝑖Log(1+𝑖𝑧1βˆ’π‘–π‘§)

In conclusion, the complex logarithm is a generalization of the real logarithm that is a multi-valued function. Its formula, log(z)=ln|z|+iarg(z)log z equals l n the absolute value of z end-absolute-value plus i arg z

log(𝑧)=ln|𝑧|+𝑖arg(𝑧)

, reveals its fundamental connection to the polar form of a complex number. To make it a single-valued function, a "branch" is selected by restricting the argument, leading to branch cuts and branch points. Understanding this function is vital for deeper work in complex analysis and related fields.

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