Solving Natural Log Equations: Tips & Tricks

[ahmadmz]

Homework Statement

sqrt(ln(x)) = ln(sqrt(x))

Homework Equations

The Attempt at a Solution

I've been trying to do this for some time now. Could anyone give some tips on how to get started with this?

 

[physicsnoob93]

Try squaring both sides and then bring the ln over.

 

[ahmadmz]

(ln(sqrt(x))^2 - ln x = 0

2 * ln(sqrt(x)) - ln x = 0

ln (sqrt(x)) (2-ln(sqrt(x))) = 0

x=1 and e^4.

Ok got it now. I tried this for 10+ minutes and was totally lost. Somehow brain starts to work a little after posting here :P

 

[physicsnoob93]

Haha yeah happens sometimes.

Btw, what do u mean by:
2 * ln(sqrt(x)) - ln x = 0

this is what i did:

After squaring:
ln(x) = ln (x^1/2)*ln(x^1/2)
Bring the half down:
ln(x) = 1/2(ln(x))*1/2(ln(x))
then after a bit of rearranging:
1/4(ln(x)ln(x)) - ln(x) = 0.
Factor the ln(x) out.
ln(x)[1/4(lnx) -1] = 0.
Solutions:
ln x = 0 or 1/4lnx = 1
x = e^0
=1 or lnx = 4, x = e^4

 

[snipez90]

(ln(sqrt(x))^2 - ln x = 0

2 * ln(sqrt(x)) - ln x = 0

ln (sqrt(x)) (2-ln(sqrt(x))) = 0

x=1 and e^4.

Ok got it now. I tried this for 10+ minutes and was totally lost. Somehow brain starts to work a little after posting here :P

After the first line above, the rest of your manipulations don't seem valid. [ln(sqrt(x))]^2 does NOT equal 2 * ln(sqrt(x)). 2 * ln(sqrt(x)) = ln[(sqrt(x))^2]. See the difference?

Similarly ln(sqrt(x)) * ln(sqrt(x)) does NOT equal ln(x). Only the addition of two logs with the same base allows for the arguments to be multiplied together. I think you need to check your rules again.

 

[HallsofIvy]

$$\sqrt{ln(x)}= ln(\sqrt{x})$$

I think I would have been inclined to write this as
$$\sqrt{ln(x)}= (1/2)ln(x)$$
and let u= ln(x) so I have
$$\sqrt{u}= (1/2)u$$
or u= u²/4. Then, then, is equivalent to u²- 4u= 0 which has u= 0 and u= 4 as solutions. Since u= ln(x), u= 0 gives x= 1 and u= 4 gives u= e⁴.

 

[ahmadmz]

Yea i made a mistake there. I was fixing it yesterday but it said can't edit after 1 hour. This is how I did it.

square both sides and factor ln(sqrt(x)) out.

ln x = (ln sqrt(x))^2

ln(sqrt(x)) (ln(sqrt(x)) - 2) = 0

ln(sqrt(x)) = 0 and ln(sqrt(x)) - 2 = 0

x=1 x=e^4

It's good to see different ways of doing it :)
Thanks for the help, please tell me if there's any mistake in this.