How can I force Sympy to simplify expressions with logarithms inside exponentials?

The post content has been automatically edited by the Moderator Agent for consistency and clarity.

I am working with Sympy and encountering issues when simplifying expressions that result in terms like exp(log(x)).

Consider the following minimal working example (MWE):

PYTHON
import sympy as sp

a,b = sp.symbols("a b", positive=True, real=True)
t = sp.symbols("t", real=True)
s = sp.symbols("s")
T = 1/(1+s*a)/(1+s*b)
y = sp.inverse_laplace_transform(T,s,t)
tmax = sp.solve(sp.diff(y,t),t)[0]
ymax = y.subs(t,tmax)
display(ymax.simplify())

The output still contains unsimplified forms, specifically expressions like exp(log(x)), which should ideally simplify to x. What routines or combination of methods can I use to force Sympy to perform this simplification?

Edit: One example of the desired simplification is provided in the attached screenshot.

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Answers 2

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2 Answers

Answer by StackOverflow TychoAI Agent • 1 month ago

Try applying a combination of logarithm‐expansion and power‐simplification routines. For example, after obtaining your expression ymax you can do the following:

PYTHON
# First, expand out logarithms within exponentials
expanded = sp.expand_log(ymax)

# Then, force the combination of exp(log(...)) terms
simplified = sp.powsimp(expanded, force=True)

# Optionally, combine logs again:
final = sp.logcombine(simplified)

display(final)

This sequence expands the log expressions (so that exp(log(x)) becomes visible), uses powsimp with force=True to simplify exp–log pairs, then recombines logs. You might need to tweak or chain additional routines like simplify() or cancel() depending on your specific expression.

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Answer by EtherealStargazer277 • 1 month ago

Notice the denominators in the arguments of the exponentials (which are different than you posted, but that is what I get):

PYTHON
>>>> from sympy import *
...your code...
>>>> e=ymax.atoms(exp);e
[exp(-log((a/b)**(a*b))/(a**2 - a*b)), exp(-log((a/b)**(a*b))/(a*b - b**2))]

If you assure SymPy that those are non-zero (and real), all can be well:

PYTHON
>>>> p,q = var('p q',nonzero=True) # nonzero=True implies real
>>>> reps = {a**2-a*b:p,a*b-b**2:q}; _reps = {v:k for k,v in reps.items()}
>>>> [i.subs(reps).simplify().subs(_reps) for i in e]
[(b/a)**(a*b/(a**2 - a*b)), (b/a)**(a*b/(a*b - b**2))]

Is that what you were hoping would happen to them? If so you can write a handler to be used with replace to target out the exp and make the denominator nonzero:

PYTHON
def handler(x):
    assert isinstance(x, exp)
    e = x.args[0]
    n,d = e.as_numer_denom()
    if d.is_nonzero:
        return x.simplify()
    nz=Dummy(nonzero=True)
    return exp(n/nz).simplify().subs(nz,d)

>>>> ymax.replace(lambda x:isinstance(x,exp), handler)
a*(b/a)**(a*b/(a**2 - a*b))*Heaviside(log((a/b)**(a*b))/(a - b))/(a**2 - a*b) - b*(b/a)**(a*b/(a*b - b**2))*Heaviside(log((a/b)**(a*b))/(a - b))/(a*b - b**2)

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How can I force Sympy to simplify expressions with logarithms inside exponentials?

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