Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm getting stuck trying to solve this logarithmic equation:

$$ \log( \sqrt{4-x} ) - \log( \sqrt{x+3} ) = \log(x) $$ I understand that the first and second terms can be combined & the logarithms share the same base so one-to-one properties apply and I get to: $$ x = \frac{\sqrt{4-x}}{ \sqrt{x+3} } $$ Now if I square both sides to remove the radicals: $$ x^2 = \frac{4-x}{x+3} $$ Then: $$ x^2(x+3) = 4-x $$ $$ x^3 +3x^2 + x - 4 = 0 $$

Is this correct so far? How do I solve for x from here?

share|improve this question
1  
There are formulas for solving cubic equations, but there does not seem to be any "nice" solution in this case. –  Per Manne Oct 17 '12 at 16:52
    
Yes, very good! Can you check the exercise again? Wasn't there a $2$ or $\sqrt x$ somewhere? Wolfram Alpha says, has one real solution but ugly: $x=0.893289..$ –  Berci Oct 17 '12 at 16:53
    
@Berci Thanks, I copied it correctly. It looks like they are looking for the ugly solution! –  Justin Brown Oct 17 '12 at 17:11
add comment

2 Answers

up vote 2 down vote accepted

Fine so far. I would just use Wolfram Alpha, which shows there is a root about $0.89329$. The exact value is a real mess. I tried the rational root theorem, which failed. If I didn't have Alpha, I would go for a numeric solution. You can see there is a solution in $(0,1)$ because the left side is $-4$ at $0$ and $+1$ at $1.$

share|improve this answer
    
Okay, I put the equation in Wolfram Alpha and got the same root, thank you. Do you have a link that explains how to get a numeric solution? –  Justin Brown Oct 17 '12 at 17:14
1  
@JustinBrown: There are many methods, discussed in numerical analysis texts. The simplest to explain is bisection. We know that $f(0) \lt 0$ and $f(1) \gt 0$, so there is a root in there. Check $f(0.5)=-2.625$ and we know the root is in $(0.5,1)$, then check $f(0.75)$ and so on. Keep going until the interval is small enough. There are other methods that converge faster. –  Ross Millikan Oct 17 '12 at 17:22
    
I'm familiar with that technique I just didn't realize that is what you meant. Thank you! –  Justin Brown Oct 17 '12 at 17:31
add comment

It is correct so far.

There is clearly a root between $0$ and $1$. Either use numerical methods to find it is about $0.893289$ or (not recommended) solve the cubic to get $$\sqrt[3]{\frac{3}{2} - \sqrt{\frac{211}{108}}} + \sqrt[3]{\frac{3}{2} + \sqrt{\frac{211}{108}}} -1$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.