Logarithmic equations over R




Difficulties when solving logarithmic equations.

On this page log is the logarithm with base 10. If the logarithm has another base than 10, the method to solve the equation does not change.

Explanation by means of an example.

 
log(x2 - 2) = log (-x)
The general method to solve an equation consists in repeatedly replacing the equation by an equivalent equation.

To solve the equation, we are tempted to omit the log-function. But the following equivalence is false!

 
log(x2 - 2) = log (-x)   <=> x2 - 2 = -x
Notice that 1 is a solution of x2 - 2 = -x but it is not a solution of log(x2 - 2) = log (-x). So, the equations log(x2 - 2) = log (-x) and x2 - 2 = -x are not equivalent.

It is clear that following expression is correct.

 
log(x2 - 2) = log (-x)   =>  x2 - 2 = -x
If two numbers have the same logarithm, then those numbers are equal. All solutions of the equation on the left are solutions of the equation on the right but not vice versa.

But we will solve logarithmic equations by omitting the log-function on both sides. But every time we must be aware that, by omitting the log-function, the new equation can have more solutions than the previous one. We really do not want these extra solutions. We call them parasitic solutions.

There are several ways to eliminate the parasitic solutions. The easiest way is testing the solutions to the given equation. The solutions that do not satisfy the original equation are the parasitic solutions. They are deleted.

Let's use this procedure to the equation log (x2 - 2) = log (-x).

 
log(x2 - 2) = log (-x)   =>  x2 - 2 = -x
This gives a quadratic equation with roots 1 and -2. We test these two values to the original equation and we see that only -2 is a solution. The value 1 is a parasitic solution and must be deleted.

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