[java-logging-logback] Full stack in case of an exception not logged

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Please run down the following list and make sure you've tried the usual "quick fixes":

- Search the issues already opened: https://github.com/googleapis/java-logging-logback/issues
- Check for answers on StackOverflow: http://stackoverflow.com/questions/tagged/google-cloud-platform

If you are still having issues, please include as much information as possible:

#### Environment details

1. Specify the API at the beginning of the title. For example, "BigQuery: ...").
General
2. OS type and version: All
3. Java version: 17
4. version(s): 0.123.8-alpha

#### Steps to reproduce

1. Log any exception - `log.error("a message", e);
2. The message showing up in the console or in stack driver has only the top level message from the stack trace. The remaining stack is lost. The approach taken by logback's console appender is to print the remain stack in a different field called "exception" this way keeping it separate from the main message which also works.

#### Code example

```java
// example
```

#### Stack trace
```
logger.error("some exception message", e);
```

```
{"severity":"ERROR","timestamp":"2022-04-30T03:18:00.315Z","logging.googleapis.com/labels":{"levelName":"ERROR","loggerName":"org.springframework.boot.autoconfigure.web.reactive.error.AbstractErrorWebExceptionHandler","levelValue":"40000"},"logging.googleapis.com/trace_sampled":false,"@type":"type.googleapis.com/google.devtools.clouderrorreporting.v1beta1.ReportedErrorEvent","message":"[d1dbed05-3] 500 Server Error for HTTP GET \"/greetings/exception\"\njava.lang.RuntimeException: Throwing an explicit exception\n at SomeClass.greetingsWithException(GreetingsController.kt:27)"}
```

#### External references such as API reference guides

#### Any additional information below

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