Which database transaction log will increase?
Creating a minimal, complete, and verifiable example will help you better formulate questions in future, and allow other folks to help you more easily.

I've created one:

CREATE DATABASE db1;
CREATE DATABASE db2;
GO
USE db1;
CREATE TABLE dbo.table1 (id1 int, id2 int);
USE db2
CREATE TABLE dbo.table1 (id1 int, id2 int);
CREATE TABLE dbo.table2 (id1 int, id2 int);

INSERT INTO db1.dbo.table1(id1, id2) VALUES (1, 2);
INSERT INTO db1.dbo.table1(id1, id2) VALUES (3, 2);
INSERT INTO db2.dbo.table1(id1, id2) VALUES (1, 2);
INSERT INTO db2.dbo.table2(id1, id2) VALUES (1, 2);

DELETE FROM db1.dbo.table1
FROM dbo.table1 t1
    INNER JOIN dbo.table2 ON t1.id1 = table2.id1


Looking at the tables is instructive:

SELECT *
FROM db1.dbo.table1;



╔═════╦═════╗
║ id1 ║ id2 ║
╚═════╩═════╝

SELECT *
FROM dbo.table1;



╔═════╦═════╗
║ id1 ║ id2 ║
╠═════╬═════╣
║   1 ║   2 ║
╚═════╩═════╝

SELECT *
FROM dbo.table2;



╔═════╦═════╗
║ id1 ║ id2 ║
╠═════╬═════╣
║   1 ║   2 ║
╚═════╩═════╝

Cleanup:

USE master;
DROP DATABASE db2;
DROP DATABASE DB1;


The takeaway appears to be that all rows in db1.dbo.table1 are deleted, which makes sense since the FROM dbo.table1 JOIN dbo.table2 ... clause does not reference database1.dbo.table1 at all.