A Non-standard Exponential Equation

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[davidblauyoutube]

Here's my take on the problem:
1. Rewrite the equation as 1/10 = x^9 e^(-x ln 10), then take 9th roots to yield c = x e^(x ln c) where c = 10^(-1/9).
2. Multiply both sides by (ln c) to get c ln c = (x ln c) e^(x ln c).
3. Apply the Lambert W function to get W(c ln c) = x ln c.
4. Solve as x = W(c ln c) / ln c. The principle branch yields W0(c ln c) = ln c and x = 1, while the other real branch yields x= W(-1, c ln c) / ln c = 10.
There are additional complex solutions from the other branches of the W function, e.g. the -2 branch gives x ~ 14.5884 + 28.8696 i.

[brucelavoie]

If you replace 10 with k and 9 with b then this equation and it's results hold as long as b=k-1. The points of intersection occur at x = 1 and x = k. You switch up and use log base k in the equation to replace log base 10.

[bxyhxyh]

Ok. Here it was easy to guess. But how can we guess if equation was bit different that second solution was some irrational number?

[mehmeterciyas]

How did we find x=10? 1 is obvious but 10?

[AB-oyon]

So by inspection, it wasn't too difficult to get x=1, 10. I didnt think to use a geometrical interpretation. I couldnt think of an analytical method to show there aren't more solutions

[goldfing]

Of course it it is solvable, because, like so many of these examples, it has an integer solution, here x = 1:
10^(1 - 1) = 10^0 = 1 = 1^9. The only question is if there are any additional solutions. We can write
10^(x - 1) = 10^x / 10^1 = x^9 and mulitply by 10:
10^x = 10*x^9
Since 10 plays such a role in this equation, I try another integer solution, namely x = 10:
10^10 = 10*10^9, which is true. For x > 10, the exponential function y = 10^x grows faster than the power function y = 10*x^9. For x < 0, there is no solution because 10^x > 0 for all x in R, whilst 10*x^9 < 0 for x < 0. The only interval where a solution could be located is between 0 and 1. But in this interval, 10^x > 1 and 10*x^9 < 1, so there is no solution possible. I believe that L = {0, 1} is the complete set of solutions.

[christianthomas]

observing the equation 10^(x-1) = x^9, for me it was obvious that for x=10 we had a solution. and when x=0, 10^0 = 1 and 1^9=1. I have been always advised to look first for obvious solutions.

[frentz]

1:12 "what do we do? well .. a lot of times we use a very interesting approach in algebra .. we use a graph" :)

[bollyfan]

10^(x-1) = x^9
10^(x-1) = x^(10-1)
By symmetry, x = 10 is a solution
also x = 1 is a solution because 10^(0) = 1 = 1^9
So two solutions you get right off the bat. :)

[victorchoripapa]

So, it only has two solutions? I tried to solve myself and I did this:
F(x)=10^(x-1)-x^9
If x=1: F(1)=0
If x=10: F(10)=0
When I evaluated for other values of x, I had that if x≠1 or x≠10 F(x)≠0
But good, I needed to do graphic. Thanks teacher!

[yuvalmagen]

Thank you. Is there another way to find x=10 instead of check & guess? algebra? also have to say that I liked your rational about the 2nd derivative.

[josephsilver]

This is another way to "guess" the solution of x=10. Start with the initial equation and take the log of both sides using x as a base. (x-1)*logx(10)=9*logx(x). Using substitution from the equation developed in the video [x-1=9*log10(x)] yields log10(x)*logx(10)=1. By inspection then x can equal 10 without using the graph.

[tavakoliferi]

I prefer your method of solving this equation that contains exponents, this method is also useful for solving trigonometric equations.

[eugeneimbangyorteza]

It took me 0 second to think of 2 answers ahahhahaha

[Prashant]

Shall we apply Lambert W function to this question, plz say

[lakshay-musicalscientist]

I hav another solution of how to solve this algebraically,
(x-1) ln 10=9(ln x). (Assuming (ln 10)/9=k )
k (x-1) = ln x (e power on both sides)
(e^k) * e^(x-1)= x
(e^k)/e * (e^x) =x (Assuming (e^k)/e as another constant K😂)
K e^x =x
K= x e^-x
-K=(-x)e^(-x) (Applying Lambert W function on both sides )
W(-K)=-x
x = -W(-K) = -W ( e^ (ln 10/9 -1) )

[novidsonmychanneljustcomme]

By setting 9=10-1 I saw x=10 immediately. 🤓 Nevertheless for the 2nd solution it was worth watching the video. 😉

[frentz]

that's the secret, people .. just guess. you'll never learn to solve the equation, your only chance is to guess