PowerExpand

Expand powers and logarithms of products in expr as though every variable were positive. Provided by @enumeratio/analytic.

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PowerExpand(expr)expr with Ln and Power distributed over products, assuming positivity.

Typed signatures

Type	Package	Applies
`(expression) -> expression`	`analytic`	always

Domain: Transformations

Details

A bottom-up rewrite: Ln(xy) → Ln(x) + Ln(y), Ln(x^n) → n·Ln(x), (ab)^c → a^c·b^c, and the same two rules specialized for Sqrt.
Does not check or track positivity -- it assumes it unconditionally, same as Wolfram's default (PowerExpand ignores Assumptions unless told otherwise, which this does not model either).

Examples

Basic #

$\sqrt{x^2} = x$ for positive $x$

In

\mathrm{PowerExpand}(\sqrt{x^2})

Out

x

Splits a logarithm of a product

In

\mathrm{PowerExpand}(\ln(xy))

Out

\ln(x)+\ln(y)

Splits a product under the root into a product of roots

In

\mathrm{PowerExpand}(\sqrt{xy})

Out

\sqrt{x}\sqrt{y}

Scope #

Pulls out an exponent

In

\mathrm{PowerExpand}(\ln(x^{n}))

Out

n\ln(x)

Distributes a power over a product

In

\mathrm{PowerExpand}((ab)^{c})

Out

a^{c}b^{c}

In

\mathrm{PowerExpand}(\sqrt{ab})

Out

\sqrt{a}\sqrt{b}

A product raised to a power splits termwise

In

\mathrm{PowerExpand}((xy)^{a})

Out

x^{a}y^{a}

Takes the root of a square as the argument, ignoring branches

In

\mathrm{PowerExpand}(\sqrt{z^2})

Out

z

The log of a reciprocal gives the negative log

In

\mathrm{PowerExpand}(\ln(\frac{1}{z}))

Out

-\ln(z)

The power comes out of the log: $\ln z^a = a\ln z$

In

\mathrm{PowerExpand}(\ln(z^{a}))

Out

a\ln(z)

Possible issues #

The branch-blind rule gives $(x-y) + (y-x) = 0$

In

\mathrm{PowerExpand}(\sqrt{(x-y)^2}+\sqrt{(y-x)^2})

Out

0

About

Examples	11
Checked against	Wolfram Language 11 agree
Crosswalk	Wolfram Language

See also: FunctionExpand, Ln