You should use the Null Object Pattern. This has two advantages for you: 1) you can get your tests up and running quickly and they won't "blow up", and 2) anyone will be able to use your class without supplying a logger. Just use NullLogger.Instance, or NullLoggerFactory.Instance.
However, you should use a mocking framework to verify that log calls get made. Here is some sample code with Moq.
    [TestMethod]
    public void TestLogError()
    {
        var recordId = new Guid("0b88ae00-7889-414a-aa26-18f206470001");
        _logTest.ProcessWithException(recordId);
        _loggerMock.Verify
        (
            l => l.Log
            (
                //Check the severity level
                LogLevel.Error,
                //This may or may not be relevant to your scenario
                It.IsAny<EventId>(),
                //This is the magical Moq code that exposes internal log processing from the extension methods
                It.Is<It.IsAnyType>((state, t) =>
                    //This confirms that the correct log message was sent to the logger. {OriginalFormat} should match the value passed to the logger
                    //Note: messages should be retrieved from a service that will probably store the strings in a resource file
                    CheckValue(state, LogTest.ErrorMessage, "{OriginalFormat}") &&
                    //This confirms that an argument with a key of "recordId" was sent with the correct value
                    //In Application Insights, this will turn up in Custom Dimensions
                    CheckValue(state, recordId, nameof(recordId))
            ),
            //Confirm the exception type
            It.IsAny<NotImplementedException>(),
            //Accept any valid Func here. The Func is specified by the extension methods
            (Func<It.IsAnyType, Exception, string>)It.IsAny<object>()),
            //Make sure the message was logged the correct number of times
            Times.Exactly(1)
        );
    }
    private static bool CheckValue(object state, object expectedValue, string key)
    {
        var keyValuePairList = (IReadOnlyList<KeyValuePair<string, object>>)state;
        var actualValue = keyValuePairList.First(kvp => string.Compare(kvp.Key, key, StringComparison.Ordinal) == 0).Value;
        return expectedValue.Equals(actualValue);
    }
For more context, see this article.