Log2File is a very useful way to redirect your phone logging into a file.
You do not need to be connected to a computer to save the logs that are generated by the phone.
You can activate this application and save all the logs you want into your sdcard.

Features:
-Save the logs in /sdcard/log.txt file
-View the log at runtime (read logs)
-Share your device's system log
-Simulating adb logcat command

You can also send the logs to your email or any other way of messaging.

This app collects a device log and gives you an option to send it to a someone (usually a developer) using a method (email, messaging, etc.).
The log data is vital to troubleshoot the crashing of some application.
Also when you look the log of the phone at runtime you can feel like a hacker or at least some geek.

It is really powerful thing to have the phone log in your hands.
In this way you can see which applications are running at this moment what is happening with you memory , you can see also if there is some application that is generating a lot of logging etc.

The system log is called logcat by the developers.
When the phone is connected with computer it can be seen with adb logcat command.

This log contains many useful information regarding what the applications and the Android OS are performing at your phone.

This app is completely free and without in-app payments.

This is the free version of a Logcat.
Hava fun with this Android Terminal Emulator

Screenshots: