Simplify logarithmic expression

asked 2024-09-12 13:54:04 +0100

kaba
13 ●3

updated 2024-09-12 17:33:31 +0100

FrédéricC

5311 ●4 ●44 ●118

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.

answered 2024-09-12 14:53:56 +0100

Emmanuel Charpentier
7774 ●6 ●51 ●154

updated 2024-09-12 16:34:05 +0100

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 ( 2024-09-12 20:06:30 +0100 )edit

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 ( 2024-09-13 11:35:15 +0100 )edit

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