Simplify logarithmic expression

full_simplify

asked 0 years ago

kaba
13 ●3

updated 0 years ago

FrédéricC

5426 ●4 ●44 ●121

I am trying to simplify the expression

1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))

under the assumptions that x is real, and x > 1. I expect

-1/2 * acosh(x)

I tried

with assuming(x, 'real', x > 1):
    print((1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))).full_simplify())

However, that just prints back the original expression.

I would do this by hand as follows:

 1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
 = 1/2 * log(2) - 1/2 * log(2 * (x + sqrt(x^2 - 1)))
 = 1/2 * log(2) - 1/2 * log(2) - 1/2 * log(x + sqrt(x^2 - 1))
 = -1/2 * log(x + sqrt(x^2 - 1))
 = -1/2 * acosh(x)

How can I get SageMath to perform this simplification?

Even without the acosh simplification I would expect to get rid of the 1/2 * log(2) term.

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1 Answer

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answered 0 years ago

Emmanuel Charpentier
7804 ●8 ●52 ●155

updated 0 years ago

Sage's simplify{_xxx}methods are known to be somewhat weaker than what is available in other interpreters. Try :

sage: (1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1)))._sympy_().simplify()._sage_()
-1/2*log(x + sqrt(x^2 - 1))
sage: (1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1)))._giac_().simplify()._sage_()
1/2*log(x - sqrt(x^2 - 1))
sage: (1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1)))._mathematica_().FullSimplify().sage()
-1/2*log(x + sqrt(x^2 - 1))

The acosh reexpression seems, however, unavailable in any of the interpreters I tried. Extending the current demoivre and exponentialize methods to inverse trigonometric and hyperbolic functions may be possible.

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Comments

Yes, that seems to at least take care of the logarithm. Hopefully SageMath develops better simplification capability later.

kaba ( 0 years ago )

An alternative command for the solution proposed by Emmanuel is

sage: s = 1/2 * log(2) - 1/2 * log(2 * x + 2 * sqrt(x^2 - 1))
sage: simplify(s, algorithm='sympy')
-1/2*log(x + sqrt(x^2 - 1))

eric_g ( 0 years ago )

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Last updated: Sep 12 '24

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