Here is how I got cloud logging to work using SLF4J.  This works on a non-compatible Java GAE Flex environment.
logback.xml 
<configuration debug="true">
    <appender name="FILE" class="ch.qos.logback.core.FileAppender">
        <file>/var/log/app_engine/custom_logs/app.log.json</file>
        <append>true</append>
        <!-- encoders are assigned the type ch.qos.logback.classic.encoder.PatternLayoutEncoder by default -->
        <encoder class="ch.qos.logback.core.encoder.LayoutWrappingEncoder">
            <layout
                class="putyourpackagenamehere.GCPCloudLoggingJSONLayout">
                <pattern>%-4relative [%thread] %-5level %logger{35} - %msg</pattern>
            </layout>
        </encoder>
    </appender>
    <root level="DEBUG">
        <appender-ref ref="FILE" />
    </root>
</configuration>
Here is the PatternLayout class I used to produce the JSON on a single line in the log file.
import static ch.qos.logback.classic.Level.DEBUG_INT;
import static ch.qos.logback.classic.Level.ERROR_INT;
import static ch.qos.logback.classic.Level.INFO_INT;
import static ch.qos.logback.classic.Level.TRACE_INT;
import static ch.qos.logback.classic.Level.WARN_INT;
import java.util.Map;
import org.json.JSONObject;
import com.homedepot.ta.wh.common.logging.GCPCloudLoggingJSONLayout.GCPCloudLoggingEvent.GCPCloudLoggingTimestamp;
import ch.qos.logback.classic.Level;
import ch.qos.logback.classic.PatternLayout;
import ch.qos.logback.classic.spi.ILoggingEvent;
/**
 * Format a LoggingEvent as a single line JSON object  
 * 
 *  <br>https://cloud.google.com/appengine/docs/flexible/java/writing-application-logs
 *  
 *  <br>From https://cloud.google.com/appengine/articles/logging
 *  <quote>
 *  Applications using the flexible environment should write custom log files to the VM's log directory at 
 *  /var/log/app_engine/custom_logs. These files are automatically collected and made available in the Logs Viewer. 
 *  Custom log files must have the suffix .log or .log.json. If the suffix is .log.json, the logs must be in JSON 
 *  format with one JSON object per line. If the suffix is .log, log entries are treated as plain text.
 *  </quote>
 *  
 *  Nathan: I can't find a reference to this format on the google pages but I do remember getting the format from some
 *  GO code that a googler on the community slack channel referred me to.   
 */
public class GCPCloudLoggingJSONLayout extends PatternLayout {
    @Override
    public String doLayout(ILoggingEvent event) {
        String formattedMessage = super.doLayout(event);
        return doLayout_internal(formattedMessage, event);
    }
    /* for testing without having to deal wth the complexity of super.doLayout() 
     * Uses formattedMessage instead of event.getMessage() */
    String doLayout_internal(String formattedMessage, ILoggingEvent event) {
        GCPCloudLoggingEvent gcpLogEvent = new GCPCloudLoggingEvent(formattedMessage
                                                                    , convertTimestampToGCPLogTimestamp(event.getTimeStamp())
                                                                    , mapLevelToGCPLevel(event.getLevel())
                                                                    , null);
        JSONObject jsonObj = new JSONObject(gcpLogEvent);
        /* Add a newline so that each JSON log entry is on its own line.
         * Note that it is also important that the JSON log entry does not span multiple lines.
         */
        return jsonObj.toString() + "\n"; 
    }
    static GCPCloudLoggingTimestamp convertTimestampToGCPLogTimestamp(long millisSinceEpoch) {
        int nanos = ((int) (millisSinceEpoch % 1000)) * 1_000_000; // strip out just the milliseconds and convert to nanoseconds
        long seconds = millisSinceEpoch / 1000L; // remove the milliseconds
        return new GCPCloudLoggingTimestamp(seconds, nanos);
    }
    static String mapLevelToGCPLevel(Level level) {
        switch (level.toInt()) {
        case TRACE_INT:
            return "TRACE";
        case DEBUG_INT:
            return "DEBUG";
        case INFO_INT:
            return "INFO";
        case WARN_INT:
            return "WARN";
        case ERROR_INT:
            return "ERROR";
        default:
            return null; /* This should map to no level in GCP Cloud Logging */
        }
    }
    /* Must be public for JSON marshalling logic */
    public static class GCPCloudLoggingEvent {
        private String message;
        private GCPCloudLoggingTimestamp timestamp;
        private String traceId;
        private String severity;
        public GCPCloudLoggingEvent(String message, GCPCloudLoggingTimestamp timestamp, String severity,
                String traceId) {
            super();
            this.message = message;
            this.timestamp = timestamp;
            this.traceId = traceId;
            this.severity = severity;
        }
        public String getMessage() {
            return message;
        }
        public void setMessage(String message) {
            this.message = message;
        }
        public GCPCloudLoggingTimestamp getTimestamp() {
            return timestamp;
        }
        public void setTimestamp(GCPCloudLoggingTimestamp timestamp) {
            this.timestamp = timestamp;
        }
        public String getTraceId() {
            return traceId;
        }
        public void setTraceId(String traceId) {
            this.traceId = traceId;
        }
        public String getSeverity() {
            return severity;
        }
        public void setSeverity(String severity) {
            this.severity = severity;
        }
        /* Must be public for JSON marshalling logic */
        public static class GCPCloudLoggingTimestamp {
            private long seconds;
            private int nanos;
            public GCPCloudLoggingTimestamp(long seconds, int nanos) {
                super();
                this.seconds = seconds;
                this.nanos = nanos;
            }
            public long getSeconds() {
                return seconds;
            }
            public void setSeconds(long seconds) {
                this.seconds = seconds;
            }
            public int getNanos() {
                return nanos;
            }
            public void setNanos(int nanos) {
                this.nanos = nanos;
            }
        }       
    }
    @Override
    public Map<String, String> getDefaultConverterMap() {
        return PatternLayout.defaultConverterMap;
    }   
}