Q: How to publish a book review in a journal?

Most academic journals publish book reviews. The aim of book reviews that journals publish is to provide insight and opinion on recently published scholarly books. Thus, if the book review that you have written is about an academic book, you should be able to get it published in a journal. Book reviews are a good publication option for early-career researchers as it allows the researcher to stay abreast of new literature in the field, while at the same time, adding to his/her publication list. That is probably the reason why your professor wants you to get it published in a  journal.

You should look up some of the popular journals in your field and see if they publish book reviews. Once you have selected a journal, you can send a pre-submission inquiry to the editor, mentioning which book you have reviewed and why you think the book review would be of interest to the journal's audience. However, don't send your article to the journal at this stage. If the journal editor shows interest in your book review, you can submit it to the journal using the online submission system.  

Related reading:

• How to write a book review
• "Book reviews perform an important function for authors, readers, and the discipline as a whole."
• How do journals evaluate book reviews: Perspectives from a book review editor