### isabelle

#### Is there a lemma like “∃x. a^x = b” proved in Isabelle?

Does anyone know where a lemma similar to ∃(x::real). a^x = (b::real) might be found? I couldn't find something like that in the 'query', but it seems pretty handy.

You need a few more assumptions on a and b and you need to use the powr operator instead of ^, since ^ is only for the n-th power where n is a natural number. powr on the other hand is for any non-negative real number raised to the power of any other real number. (or similarly for complex numbers) lemma fixes a b :: real assumes "a > 0" "a ≠ 1" "b > 0" shows "∃x. a powr x = b" proof from assms show "a powr (log a b) = b" by simp qed

### Related Links

Substitution in Isabelle

Converting free variables to bound variables

Isabelle type unification/inference error

Substituting for the lambda expression in Isabelle

Apply simplifier to arbitrary term

need a definition in Isabelle to show that two partial functions never produce the same output

Why must Isabelle functions have at least one argument?

Working with generic definitions in Isabelle

completely replace the inner syntax in isar?

Applying lemma to solve goal

How to get the value of a const with ML code in Isabelle?

Expressing a simple declarative proof about exponents in Isar

How to make Isabelle use ZF?

How to prove “(∀x. P) ∧ Q ⟹ ∀x. P” using conjunct1 in Isabelle?

Simple lemma in Isabelle

Integration in Isabelle