Thread: solving for an exponent

I'm having trouble with a simple math problem that I embarrassingly should know how to solve.

X^n = n^10

X is a constant value.
Solve for n.

Can anyone point me in the right direction?

Thanks!

Originally Posted by wonder-wookie

I'm having trouble with a simple math problem that I embarrassingly should know how to solve.

X^n = n^10

X is a constant value.
Solve for n.

Can anyone point me in the right direction?

Thanks!

Take the log of both sides (it doesn't matter which base you use, base 10 is convenient). Now use a well known log rule to get a linear equation in n. Solve this equation.

ok, lets see if I have this correct...

log(X^n) = log(n^10)
n log (X) = 10 log (n)

n 10
----- = ------
log (n) log (X)
That is as far as I got. How can I simplify the left hand side so there is a single N term?

Originally Posted by wonder-wookie

ok, lets see if I have this correct...

log(X^n) = log(n^10)
n log (X) = 10 log (n)

n 10
----- = ------
log (n) log (X)
That is as far as I got. How can I simplify the left hand side so there is a single N term?

Sorry, I misread the question.

In fact, the only way you can solve for n algebraically is to use the Lambert W-function (a search of these forums will give several hits).

Why are you trying to find n? Where has the question come from?

Ultimately I am trying to find an integer value of N when X^n > N^10. I was working on the assumption that if I found the point where they become equal it would be easy to just take the next larger integer to solve the problem. The values for X are greater than 1 but still small, on the order of 1.001