Is that posible to export Task Activity or Error Logs with business rule?

Hi, I am not aware of an existing Api function that allows you to do that. If no one else knows one, you will have to pull the information from the corresponding tables and join them together.

Alternatively, you can extract both from the Systems tab if that suffices:

Solved: Extract_Data_System - OneStream Community (onestreamsoftware.com)

Depending on what you want to do with it, one can e.g. load the XML into Excel to filter the information in there. 
In Excel: Data >> Get Data >> From File >> From XML

The MarketPlace solution Admin Solution Tools (AST) has examples in Dashboard Extender (log by date, by user etc):

''' <summary>
''' Performs a SQL query to retrieve a Log Analysis based on a date range
''' </summary>
''' <param name="si"></param>
''' <param name="dateRange"></param>
''' <returns></returns>
Private Function GetLogAnalysisByDate(ByVal si As SessionInfo, ByVal dateRange As Integer) As DataTable
    Try
        'Define the SQL Statement				
        Dim parameters As List(Of DbParamInfo) = New List(Of DbParamInfo) From {
                New DbParamInfo("@dateRange", dateRange)
                }

        Dim newSql As String = "
        DECLARE @LogAnalysisByDate TABLE (DaysOld VARCHAR(50), LogonCount BIGINT, ErrorCount BIGINT, TaskCount BIGINT)
        -- Set the interval in days
        DECLARE @DateInterval INT = @dateRange
        --DECLARE @DaysOld INT = @DateInterval
        DECLARE @CurrentDate DATETIME = GETUTCDATE()
        --Set start date to number of days behind current date
        DECLARE @StartDate DATETIME = DATEADD(d, (@DateInterval) * -1, GETUTCDATE())
        DECLARE @EndDate DATETIME = GETUTCDATE()
        DECLARE @TotalLogs BIGINT = (SELECT ((SELECT COUNT(*) FROM UserLogonActivity)+(SELECT COUNT(*) FROM ErrorLog)+(SELECT COUNT(*) FROM TaskActivity)))	
        DECLARE @Loop INT = 0
        DECLARE @DaysFrom INT = 0
        DECLARE @DaysTo INT = @DateInterval
        DECLARE @LogonCount INT 
        DECLARE @ErrorCount INT 
        DECLARE @TaskActivityCount INT 

        WHILE (SELECT ISNULL(SUM(LogonCount) + SUM(ErrorCount) + SUM(TaskCount), 0) FROM @LogAnalysisByDate) < @TotalLogs	
            BEGIN 			
                SET @LogonCount = (SELECT COUNT(*) FROM UserLogonActivity WHERE LogonTime BETWEEN @StartDate AND @EndDate) 
                SET @ErrorCount = (SELECT COUNT(*) FROM ErrorLog WHERE ErrorTime BETWEEN @StartDate AND @EndDate) 
                SET @TaskActivityCount = (SELECT COUNT(*) FROM TaskActivity WHERE QueuedTime BETWEEN @StartDate AND @EndDate) 		
                IF @Loop > 0 
                    BEGIN
                        -- Set the DaysOld column value
                        SET @DaysTo = @DaysTo + @DateInterval
                        SET @DaysFrom = @DaysTo - @DateInterval + 1					
                    END				

                -- Update the start and end dates
                SET @EndDate = @StartDate 
                SET @StartDate = DATEADD(d, (@DaysTo + @DateInterval) * -1, GETUTCDATE());

                -- Insert the row
                INSERT INTO @LogAnalysisByDate 
                SELECT (CONVERT(VARCHAR(50), @DaysFrom) + ' - ' + CONVERT(VARCHAR(50), @DaysTo)), @LogonCount, @ErrorCount, @TaskActivityCount 		
                -- Increment the loop count
                SET @Loop = @Loop + 1
                
            END 

            SELECT * FROM @LogAnalysisByDate "					

        'Return the DataTable
        Using DbConnFW As DbConnInfo = BRApi.Database.CreateFrameworkDbConnInfo(si)
            Return BRApi.Database.ExecuteSql(DbConnFW, newSql, parameters, False)
        End Using

    Catch ex As Exception
        Throw ErrorHandler.LogWrite(si, New XFException(si, "Unhandled Exception in GetLogAnalysisByDate() function.", ex.Message, ex.InnerException))
    End Try
End Function