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Question 201509: ln(4x)=ln(x-5). Solve, can you please help me?
Found 3 solutions by solver91311, Theo, ikleyn:
Answer by solver91311(24713)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! this can only be true if 4x = x-5.
solve for x and you have the answer.
subtract x from both sides of the equation to get
4x - x = -5
simplify to get
3x = -5
divide both sides of the equation to get
x = -5/3
that's your answer.
substitute in the equation to get
ln (4*(-5/3) = ln (-5/3 - 5)
simplify to get
ln (-20/3) = ln (-5/3 - 15/3)
simplify further to get
ln (-20/3) = ln (-20/3)
since this is an identity, it proves your answer is correct.
the problem I see here is that you are trying to take the natural log of a negative number.
all logs have to greater than 0.
negative numbers or 0 are not allowed.
if I have done this correctly, then the value of x is correct and the natural log cannot be found.
I used the calculator to try to get the answer but the calculator said the logarithm can't be found.
to prove the logic is correct, I'll give you one that can be found.
take ln (4x) = ln (x + 15)
this can only be true if 4x = x + 15
this is true if x = 5 because 4 * 5 = 5 + 15.
your answer would be x = 5 and you would have ln (20) = ln (20) which is an identify again only this time it can be solved.
Answer by ikleyn(52915)  (Show Source):
You can put this solution on YOUR website! .
ln(4x)=ln(x-5). Solve, can you please help me?
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It is one of the first problems solved by tutor @Theo at this forum.
As I read his solution, it seems unsatisfactory to me.
So, I place here my solution in a form as it should be done/presented for school students.
Since the logarithms in both sides of the given equation are equal,
the expressions (arguments) under logarithms are equal.
It gives us this equation
4x = x - 5.
Simplify and find 'x'
4x - x = -5,
3x = -5,
x = - 5/3.
But if you substitute x = -5/3 into the original equation, you will get
a negative number as the argument of the logarithm.
Logarithmic function is not defined for negative numbers - so, we conclude that
the given equation does not have solutions in real numbers. <<<---=== ANSWER
Solved.
A proper solution to a mathematical problem does not require a myriad of words.
It should be straightforward, transparent, clear and understandable, and,
in any case, it should not contain statements that mutually contradict to each other.
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