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The homepage will likely always be the page on your website with highest domain authority. While there are situations where this might not be the case, the homepage is likely the be the page that has the most backlinks (don't internally and externally). That being said, I don't think creating local landing pages for multiple locations would harm the keyword rankings for your homepage. Instead, this provides an opportunity for your client to dominate the search engine results pages and take up more than one of the listings. As they build out these localized pages, I would spend time focusing on both the geo-targeted keywords, but also building authority for these pages. Are there any local organizations, associations, teams, etc. that would link back to the localized service page? As Julie said, your homepage is going to do the most work for you. But utilize these landing pages to enhance the results!