Does having a big table that I do not query cause the whole DB to be slower?
In my experience, a big table should not slow down the whole database. it can slow down the queries on same table though but that too will depend on the structure of table and indexes.

You dont have to store the PDF in database, I'd recommend that you eliminate the BLOB column and make it a CHAR. then store the PDF files on disk and their names into database. Here is how I'd have done this. My table would be:

CREATE TABLE logs (
    id INT UNSIGNED AUTO_INCREMENT PRIMARY KEY,
    file_name CHAR(64) UNIQUE NOT NULL,
    created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
    INDEX USING BTREE (file_name)
) ENGINE = INNODB;


and then create a SHA256 sum of PDF file and use it as filename.  

Some obvious benefits:


This table is a fixed width table (See this), hence faster in all ways.
file_name column is a SHA256 hash of file which is always 64 characters in length and unique
You have option to save PDF files in any directory on disk and reconstruct the path from this data, for example if you store the files at /home/username/html/storage/log_reports/pdf and categorise them by month or week of creation you have the created_at timestamp which you can use.


so splitting the data by days of month in a given year is  

/home/username/html/storage/log_reports/pdf
    |
    |-- 2014
    |    |
    |    |-- May
    |        |
    |        |-- 5th
    |             |-- 4355a46b19d348dc2f57c046f8ef63d4538ebb936000f3c9ee954a27460dd865.pdf
    |             |-- 53c234e5e8472b6ac51c1ae1cab3fe06fad053beb8ebfd8977b010655bfdd3c3.pdf


and so on.

Example case:

CREATE TABLE logs (
    id INT UNSIGNED AUTO_INCREMENT PRIMARY KEY,
    file_name CHAR(64) UNIQUE NOT NULL,
    created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
    INDEX USING BTREE (file_name)
) ENGINE = INNODB;


Insert some random data

INSERT INTO 
  logs (file_name, created_at)
VALUES
  ('4355a46b19d348dc2f57c046f8ef63d4538ebb936000f3c9ee954a27460dd865', TIMESTAMP('2014-5-10')),
  ('53c234e5e8472b6ac51c1ae1cab3fe06fad053beb8ebfd8977b010655bfdd3c3', TIMESTAMP('2014-5-9')),
  ('1121cfccd5913f0a63fec40a6ffd44ea64f9dc135c66634ba001d10bcf4302a2', TIMESTAMP('2014-5-5')),
  ('7de1555df0c2700329e815b93b32c571c3ea54dc967b89e81ab73b9972b72d1d', TIMESTAMP('2014-5-1'));


Recreate the path to file:

SELECT CONCAT(DATE_FORMAT(created_at, '%Y/%D/%M/'), file_name, '.pdf') from logs where id = 3;


output:

2014/5th/May/1121cfccd5913f0a63fec40a6ffd44ea64f9dc135c66634ba001d10bcf4302a2.pdf


Then you can just prepend the path to storage directory and send the file for download or attachment.

Here is an example fiddle on this