How to Solve These Logarithm Equations?

[Nx2]

hi guys, I am not too sure how to do these questions:
solve for x,
a) (logx^3)^2 = logx^18
b) logx^3 + log(x^logx) = -2

so this is what i got so far:
a) (3logx)^2 = logx^18
9logx^2 = logx^18
18logx = logx^18
logx^18 = logx^18
... then i got stuck... i was clueless and did not know how to solve for x.

b) 3logx + (logx)(logx) = -2
logx^2 + 3logx +2 = 0
so let logx = s
s^2 +3s + 2 = 0
(s+1)(s+2)=0
s=-1 or s=-2
... then again i got stuck... I am not sure if what i did was right. our teacher gave us these questions that she has never taught us b4 and said this is ur assignment for the break... any help would be very much appreciated. thnx.

- Tu

 

[quasar987]

a) You did an illegal operation btw line 2 and line 3 when you said 9logx^2 = 18logx

What you got here is $9(logx)^2$ and not $9log(x^2)$. And while it is true that $9log(x^2)=18logx$, it is not that $9(logx)^2=18logx$

 

[quasar987]

b) You're almost there! you got the solutions s=-1 and s=-2, which translate into logx =-1 and logx=-2. Now you got to solve for x. Hint: Use the fact that $e^x$ is the inverse function of $logx$. This means that

$$e^{logx}=x$$

mmh.

 

[Nx2]

oo... ok... so now that i have that... how do i solve for x in a though?

 

[quasar987]

oo... ok... so now that i have that... how do i solve for x though?

see my last post. same trick.

 

[Nx2]

srry.. but i don't think we learned e^x

 

[quasar987]

Mmmh, did you learn about exp(x) ?

 

[Nx2]

yea i know exp(x)

 

[quasar987]

They are the same.

$$\mbox{exp}(x) = e^x$$

So you know that exp(x) is the inverse function of logx, right?

Then exp(logx)=x.

 

[Nx2]

ok.. but i still don't get how i would solve for x if i had
logx^18 = logx^18... wouldn't i just end up with 0 = 0?

 

[quasar987]

err. you don't have logx^18 = logx^18. I told you there was a mistake between line 2 and 3. All you have is

$$9(logx)^2 = (logx)^{18}$$

If you don't see where to go from there, use the same substitution as you did in b). Set s = logx... then solve for s. Then use the fact that exp(s)=x to solve for x.

 

[Nx2]

omg... that's why it wasnt working... i was like so clueless... forgot bout that... thnx a lot man. i c what u meant when u were talking bout e^x now... thnx, very much appreciated for putting up with me.

 

[quasar987]

It's all good. Don't hesistate to post if something doesn't work out!