Unable to access logger in client or server from plugin

[briandee]

Environment

Environment
Operating System: Mac OS Ventura
Node Version: v20.10.0
Nuxt Version: 3.10.0
Package Manager: npm@10.2.3

Reproduction

https://stackblitz.com/edit/github-bfnrsx-dxkhza?file=src%2Fplugins%2Flogger.client.ts,src%2Fplugins%2Flogger.server.ts,src%2Fmodule.ts,package.json,src%2FlogLevels.ts,playground%2Fnuxt.config.ts

Here is the configuration to access the module

      export default defineNuxtConfig({
        nitro: {
          serveStatic: true,
        },
      
        css: ["@/assets/css/global.css"],
      
        modules: ["@vueuse/nuxt", "cd-logger"],
      
        runtimeConfig: {
          public: {
            firstModule: {
              loggerOptions: {
                appName: "c_logger", 
                useIsoTime: false,  
                outPutOnSingleLine: true,  
                clientOptions: {
                  level: "info",
                }, 
                serverOptions: {
                  level: "info",    
                  useFormattersAndSerializers: true,
                },
              },
            },
          },
        },
      });

I attempt to access it as follows

        import axios from "axios";
        import { defineEventHandler, readBody } from "h3";

        const LOGIN_URL =
        process.env.LOGIN_URL ||
        "http://app-dev.com/api/v1/server/secure/users/account/login";

        export default defineEventHandler(async (event) => {
        try {
            try {
            console.log("Context:: - ", event);
            console.log("LOGGER:: - ", this.$logger);
            } catch (error) {
            console.error("Unexpected error thrown", error);
            }
            const loginRequest = await readBody(event);
            const response = await axios.post(LOGIN_URL, loginRequest);
            return { json: response.data, status: response.status };
        } catch (error) {
            console.error(error);
        }
        });

Describe the bug

Attempting to access the logger in server/api returns undefined but using const {$logger} = useNuxtApp() in client works. I can go to localhost:3000 after running npm run dev and load it without errors. The plugin provided used the information in the documentation so what am I missing? I thought I could simply be able to access the logger in the client using this.$logger without useNuxtApp but that does not work. I am unable to access the logger in the event handler which always returns undefined

Additional context

Unfortunately, I cannot provide the published plugin because it is in a private npm repo. I can access it with npm i cd-logger --save-prod and include it as a module without errors so it is difficult to understand why it is undefined when there are no errors. When I run 'npm run prepack' the node_modules is @nuxt/kit is included, is included in the published module. Is that expected?

Can I use addServerPlugin, and if so how do I reference the logger in the plugin?

Logs

No response

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• github-bfnrsx-dxkhza (Open reproduction)