Exponential form is a representation using exponentiation, with distinct uses for equations involving logarithms and for complex numbers.

For a positive base , the exponential form of the relation is

where and is real. It is equivalent to the logarithmic form of the same relation (Abramson 2021). For example, is the exponential form of .

The exponential form of a nonzero complex number is

where is its complex modulus and is a complex argument. By the Euler formula, this is equivalent to the polar form . The complex argument is determined only modulo (Churchill and Brown 1990).

See also

Complex Number, Euler Formula, Exponentiation, Logarithmic Form, Polar Form, Rectangular Form

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References

Abramson, J. "Logarithmic Functions." §6.3 in Algebra and Trigonometry, 2nd ed. Houston, TX: OpenStax, 2021. https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-3-logarithmic-functions.Churchill, R. V. and Brown, J. W. Complex Variables and Applications, 5th ed. New York: McGraw-Hill, 1990.

Cite this as:

Weisstein, Eric W. "Exponential Form." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExponentialForm.html

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