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Inductive Proof of commutativity of exponentiation

This is my proof:

PROOF: If $a,b \in \mathbb N$ then $a^b \times b^a = a^a b^b$
Proof: Since $a,b \in \mathbb N$ then $a,b > 0$ thus we have:
$$a^b \times b^a = a^{a + b \cdot \log_a b} \cdot b^{b + a \cdot \log_b a} = a^a b^b $$

My question is: Is this all that is needed to prove the commutativity of multiplication and exponentiation? Or is there another proof we can all agree on? I’ve seen similar statements made before, but I haven’t seen one written out in detail.

A:

The traditional proof given on Wolfram (and every calculus textbook) goes as follows.

Let $n_1 := a$ and $n_2 := b$ so we have
\begin{align*}
x &:= a^{n_1} \times b^{n_2} \\
&= a^{a + b\cdot \log_a b} \cdot b^{b + a \cdot \log_b a} \\
&= a^{a + b\cdot \log_a b} \cdot b^{b + a \cdot \log_b a} \cdot a^{a – a \cdot \log_a a} \cdot b^{b – b \cdot \log_b

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The Film
The film, like its predecessor Bombay Dreams, is based on the Rann’s Scooter story, where the main character drives a rickshaw and earns a living by transporting people through the traffic. The novel was adapted into the short film Rann (1985), which won the festival award for best documentary film of the year in the International Short Film Festival in Cannes in 1986.
The character of Rann is based on the author’s own experiences as a rickshaw man and similar characters of other Gujarati writers. The novel’s protagonist Bajirao ‘Rann’ Scooter (Pravin Tendulkar) gives the title to the movie, in which he is played by newcomer Adah Sharma. Adah Sharma was selected after auditioning with several other actresses.
The director Premji Singh has said that he wanted to depict that the character who is the title character, cannot afford to ever say no to anyone, and ends up taking on the role of a door-keeper for others. During the making of the film, one of Sharma’s make-up artists, Anil Lalwani, left the team, accusing Singh of racism.
Tendulkar’s performance as the character of Rann is compared by critics as the best performance in the movie, as he portrays a character, who, though egotistic at times, is an honest man and good-natured.

The Novel
The novel, published in 1985, was written by Javed Jabbar, who previously wrote the screenplay for Bombay Dreams. In his introduction to the book, Jabbar writes that he tried to explore the themes that Bombay Dreams had depicted and not to reproduce the film. The novel follows the character Rann as he works as a mechanic, rickshaw driver, and storekeeper.
Set in Bombay, the novel is mostly narrated from Rann’s point of view; his narration is interrupted at various instances, to show the events around him. The novel consists of three parts: its first part describes Rann’s childhood, where he is raised by two widowed sisters. The second part of the book deals with the events of Rann’s adolescence and the third part presents Rann as an adult.
Jabbar has said that his ideal reader is someone who has read the novel in the past and does not know the plot ahead of time. He has also said that Rann’s story is a character study of a Gujarati boy, who has grown up in a family of

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