I happened to come across an interesting mistake made by a maths student.
The mistake inter-mixed the principles of algebra with logarithm.
The maths question is to solve the value of x given the expression:
2+ log (5x - 1) = log 3x
The expression, after transferring the "log (5x -1)" to the right side of the "=", became
2 = log 3x - log (-5x + 1) !
Spotted the mistake done ?
Why was the "log (-5x + 1)" in that form?
The correct expression should be 2 = log 3x - log (5x - 1).
What actually went inside the student's mind was confusion between algebra and logarithm. He did not understand the concept of "logging" the (5x -1).
Log X is always a number!
Similarly log (5x - 1) is also a number.
Therefore log (5x - 1) moves as a number, same as in moving algebraic term.
If we have 2 + (x-a) = y, re-arranging the expression, gives us 2 = y - ( x-a).
The term "x-a" is taken as a whole, with change in the sign of (x -a) and not including that of the individual internal "a" and "x". This is basic algebra.
Moving log (5x - 1) is the same. Being a number, it operates equivalent to the algebraic manipulation.
The log (5x -1) is thus taken as a whole and sign change affects only the term as a whole. It does not affect the individual internal "5x" and "-1"!
Part of learning maths is following rules and principles.
The mistake made by the student was a reflection of correct algebraic change, but in the wrong sense. "Log" had converted the term into a number, and that was the mistake not captured.
Being careful with every steps taken in solving maths questions is a discipline one can treasure. This is one of the interesting part of doing maths!