What is the error in this simple logarithmic equation?

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  • Thread starter Thread starter Ali Asadullah
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SUMMARY

The equation log(5x-17) = log(4x-14) is incorrectly solved by stating x = 3. The error lies in the domain of the logarithmic functions, which are not defined for real numbers when x = 3, as both arguments (5x-17 and 4x-14) yield negative values. While x = 3 is a valid solution in the complex number field, it does not satisfy the conditions for real logarithmic functions. Therefore, there is no solution over the domain of real numbers.

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Ali Asadullah
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Can anyone please tell me what is wrong with this solution


log(5x-17) = log(4x-14)
As logarithmic functions are one-one so,
5x-17 = 4x-14
=> x =3
 
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That's a very good question Ali. The problem is the domain of the log function over the reals.

So there is no solution over the domain of real numbers, however x=3 is a solution if the domain is the complex number field. This is an interesting result in the sense that we require the existence of complex numbers in order for a solution to exist, even though the solution itself is not complex!
 
What's basically wrong is that neither of the logarithms (in their real sense) are defined for x=3 since their argument will be negative. Comes in what uart said.
 

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