# What is 5^n=80n

1. May 4, 2005

### One-D

I've got a problem with this:
5^n=80n
find n!
I can't solve out this one.
thanx for ur help, I'm not sure if it's the right place for this thread.

2. May 4, 2005

### Berislav

If $$n \in \mathbb{Z}$$, there is no such number.

3. May 4, 2005

### cronxeh

5^x - 80*x = 0
plot it
and so far ive found 2 solutions (oh and there are no solns in Z of course)

Approximate roots are:

n=0.012759
n=3.50133

Last edited: Oct 8, 2005
4. May 4, 2005

### Zurtex

Anyway the 2 exact solutions (in real numbers) are:

$$n = -\frac{\text{ProductLog} \left( - \frac{\ln 5}{80} \right)}{\ln 5}$$

and

$$n = -\frac{\text{ProductLog} \left(-1, - \frac{\ln 5}{80} \right)}{\ln 5}$$

Which works out to 100 sf.:

n = 0.01275934595849510712320293028417381446823346765193827957982622843955432250672941443435143426214944153

And:

n = 3.501327193786496727104387840526356291207393167955463046140101380079871277483867981890189846055191784

5. May 5, 2005

### cronxeh

and that is an 'analytical' solution?

6. May 8, 2005

### One-D

thanx for ur replies