Solving a Logarithmic Exponential Equation
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- Опубликовано: 6 авг 2022
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x^(1/log(x)) = 10, as tends to be the case with these things. So, 10·log(x) = 1. The rest is trivial. In general, for the logarithm in base b, log(b, x), we have that x^(1/log(b, x)) = b.
Wow! That really simplified nicely. Easier than i was expecting.
Yes!
Holy smokes, pretty sure this is the first time that I've worked the problem just like it was done in the video. Very enjoyable problem!
Glad to hear that!
The condition of x is x >0 and x 1
Using 1/log(x) = log_x(10), switching the bases, the equation becomes:
10 log(x) =1, log(x) = 1/10, x = 10^(1/10) is the only solution.
it would be much simpler say that x^(1/log(x)) = 10^(log(x)/log(x)) = 10 so 10*log(x) = 1 --> log(x) = 1/10 --> x = 10^(1/10)
why
I don’t think so
@@CriticSimon I agree, it would be simpler. It is just straightforward simplification.
Exactly, that’s way simpler.
@@angelmendez-rivera351 Sounds good!
Fantastic!!
subbing y = log x and solving 10 y = 1 for y makes the problem trivial
I like your demonstration.
Nice, I love a good log problem. ♥️
Thanks!
Very fun! You could also go the change of variable route by letting u = logx, so x = 10^u, and the LHS of the equation becomes u*(10^u)^(1/u) = 10u.
Then 10u = 1 when u = 1/10, so logx = 1/10 and x = 10^(1/10)
you always manage to exceed my expectations with your content!
Happy to hear that! 😍
log(x) isn't 0 because 1/log(x) would be undefined. Let a=1/log(x). Then (x^a)( log(x))=1 dividing by log(x) we get x^a = a. This is satisfied if x= a^(1/a).
Taking logs of both sides log(x) = (1/a)(log(a), Multiply by a gives... a times log(x) =log(a). Since a=1/log(x) we have 1=log(a) or 10^1 = a that is a=10.
So x=a^(1/a) and x=10^(1/10)
Another Masterpiece
Thank you! 💖
On a video of your channel i guess you have analyzed two cases pair or unpair in same equation can you send me the link thanks!
Got it!
Thank you for your genius problems, when I heard the sound ‘1st, 2nd, 3rd’, Oh my god, I was lucky I was not with you in high school! Haha !!!
You are welcome! 😁
The blue graph is y = 10 * log(x)
;-)
cheers, Uli
Thnku
in what application u r solving that equation?
Notability
it got me wrong when I converted X^(1/logX) to X^(-logX).
I was thinking it originally was 1/X^(logX)...
Answer x=10^1/10 or 1.2589 or 10^0.1
x^1/logx. logx =1
x^ log x 10 = 1/logx change 1/log 10 x to log x 10, then divide both sides by logx
x^log x 10 = logx 10
10 =log x 10 (n^log n p =p)
x^10= 10 ( change to exponential form)
x = 10 ^ 1/10 raised both sides to the power of 1/10 Answer
x= 1.258925 or 10^1/10 answer
thanks so much can you please do functional equtions or number theory
Np. Sure!
Use logx = y
and solve it
There are a number of ways to do it.
x^1/logx = 1/logx (bring log x to the right)
x^logx 10= log x 10 rules of log
10 = log x 10 rules of log
x^10=10
x= 10^1/10 answer
4^x+3^2y=6Can you explain the solution to this issue
1/e seems to be the only answer...presuming log(x) means ln(x)
1 + log(logx) = 0
Log x = 0,1
X = 10^0,1
Brilliant solution!
@@BabisKoutroulis thanks
It would be much easier to set x=10^n
10^1/10=1.Log1=0. Therefore we conclude that the equation is false!
Take log(x)=t and x=10^t.........then it's looks like more easy
You totally missed a trick with this one. That trick comes from logₕx=logₐx/logₐb, and thus 1/logₕx=logₐb/logₐx. If we let a=x, then we get the trick: 1/logₕx=logₓb. This means x^(1/logₕx)=x^logₓb=b. Thus, our equation goes like 1=x^(1/logₕx)*logₕx=b*logₕx ⇒ logₕx=1/b ⇒ x=b^(1/b). If the base b=10, then x=10^(1/10), which is your answer.
x=(10)^1/10
If I take the base is e
log is base 10
10^(1\10)???
My answer is e only
10^(1/10)