\[\Large \log^2x^3-20\log\sqrt x +1=? \]
How many solutions does equation have?
1
2
3
4

- anonymous

\[\Large \log^2x^3-20\log\sqrt x +1=? \]
How many solutions does equation have?
1
2
3
4

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- anonymous

I can't start it :(

- anonymous

I know that \[\Large \log^2x =2\log x\] and \[\Large \log_{x^t}a=\frac 1t \log_xa\] and something else but as much as I see I don't need the second one here ... :S Any hint??

- Mertsj

What is it equal to? If it is an equation, it must be equal to something.

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## More answers

- anonymous

sorry... equal to 0 :$ my bad

- anonymous

I thought that I write that :$

- Mertsj

\[\log_{10}x^2=2\log_{10}x \]

- Mertsj

\[(\log_{10}x )^{2}\neq2\log_{10}x \]

- anonymous

\[\Large \log^2x =\log x\cdot \log x ??\]

- Mertsj

Yes

- TuringTest

\[\Large \log^2x^3-20\log\sqrt x +1=9\log^2x-10\log x+1\]it's not equal to 0 is it?cause then it would be quadratic in log x

- TuringTest

\[\Large \log^2x^3-20\log\sqrt x +1=0\]would make this problem a little more direct

- anonymous

Is given:
\[\Large \log^2x^3-20\log\sqrt x+1=0\]

- TuringTest

sweet, then you can use the quadratic formula from where I got you too

- TuringTest

\[\log^2x^3-20\log\sqrt x +1=9\log^2x-10\log x+1=0\]substitute\[\log x=y\]

- anonymous

what about \[\Large \log \sqrt x\]

- anonymous

Ufff.. I see sorry
Let me try.. I'll see what I can do ;)

- Mertsj

Turing, sorry to be so slow, but how is log^2x^3=9log^2x

- TuringTest

not at all :)\[(\log x^3)^2=(3\log x)^2=9\log x\]

- Mertsj

Oh. I got it. Sorry.

- TuringTest

forgot the log^2 in the last part...

- Mertsj

Now it factors!!!

- TuringTest

sweet jeezus we might be able to solve it :D

- anonymous

So I have...
\[\Large x_{1/2}\frac{10\pm\sqrt{100-36}}{18}\]
And \[\Large x_{1/2}\frac{10\pm 8}{18}\longrightarrow x_1=1 \quad ,x_2=\frac19\]
Right?

- TuringTest

I don't know, I don't feel like checking
mertsj says it factors. let's see what he gets

- anonymous

ok

- TuringTest

not factorable?
ok let me see what I get...

- Mertsj

\[(9logx-1)(logx-1)=0\]

- TuringTest

I though it factored :)
so it looks like we have the right answer... (for the log part at least)

- TuringTest

I suppose to finish this out we need to know what base the log is
is it 10?

- anonymous

Yes .. because there's no base given ...

- Mertsj

So 10^(1/9) and 10

- TuringTest

\[\log x=\frac19\implies x=\sqrt[9]10\]ew... ugly number
well, ok not that ugly, but still

- TuringTest

at least the 10 is pretty :)

- anonymous

LOL ... thanks guys so much, thanks in advance

- Mertsj

Very much so. Do these answers have to be checked in the original?

- Mertsj

Thank Turing. He's the brains of the outfit.

- TuringTest

Welcome, thanks for the compliment
@Kreshnik why do you say "in advance" ?
you assume I am going to keep helping that way? lol

- TuringTest

fyi you usually say "thanks in advance" before the person has actually done the favor

- TuringTest

(English lesson, lol)

- Mertsj

Oh. It only wanted to know how many solutions there are.

- Mertsj

A true liberal arts scholar!!

- TuringTest

haha^
screw that "how may" business, once you recognize it's quadratic in logx you know there will be 2 solutions, so I guess we could have stopped there

- TuringTest

...so long as the discriminant is not zero

- anonymous

LOL ... hahaha , Since my own language it's not English , I've seen in a forum a guy who said that: Thanks in advance .... uhauhuha LOL INDEED he said that before the answer it was given, sorry if I made any mistake , I'm trying my best, I'm 18 years old, and I'm from Kosovo but I know English too (at least I think I do ! LOL ) hahaha.. @TuringTest thanks for English lessons :) ... can't stop laughing !

- TuringTest

That's cool, I'm learning Spanish and I teach English, so I know your pain

- Mertsj

Or one of the answers is not negative.

- TuringTest

good point @Mertsj

- Mertsj

You teach English??? And you're such a math whiz. Do you teach math too?

- anonymous

I know spanish too... (I understand pretty much !) but I know English better !! ... (I don't know to write Spanish at all !)

- TuringTest

I am a tutor in English and math for the school where I am studying Spanish here in Mexico
I actually suck at grammatical rules and such, but I get by being a native speaker.

- Mertsj

Be safe. And don't get involved in the drug wars!!

- TuringTest

No worries, that's a border problem.
I'm in Guadalajara, to the south-central.
Thanks though, and take care yourself :)

- Mertsj

Thanks. Will do.

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