How to Solve 5^n=80n for n?

  • Thread starter One-D
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In summary, the conversation is about solving the equation 5^n=80n and finding the value of n. The individual is having trouble solving it and asks for help. They also question if they are in the right place to ask for help. The conversation then moves on to discussing the approximate and exact solutions for n. The individual thanks for the help and asks if the second solution is considered an 'analytical' solution.
  • #1
One-D
17
0
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.
 
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  • #2
If [tex]n \in \mathbb{Z}[/tex], there is no such number.
 
  • #3
5^x - 80*x = 0
plot it
and so far I've found 2 solutions (oh and there are no solns in Z of course)

Approximate roots are:

n=0.012759
n=3.50133
 
Last edited:
  • #4
Anyway the 2 exact solutions (in real numbers) are:

[tex]n = -\frac{\text{ProductLog} \left( - \frac{\ln 5}{80} \right)}{\ln 5}[/tex]

and

[tex]n = -\frac{\text{ProductLog} \left(-1, - \frac{\ln 5}{80} \right)}{\ln 5}[/tex]

Which works out to 100 sf.:

n = 0.01275934595849510712320293028417381446823346765193827957982622843955432250672941443435143426214944153

And:

n = 3.501327193786496727104387840526356291207393167955463046140101380079871277483867981890189846055191784
 
  • #5
and that is an 'analytical' solution?
 
  • #6
thanx for ur replies
 

1. What is the first step in solving 5^n=80n?

The first step in solving this equation is to take the logarithm of both sides. This will help to isolate the variable n.

2. What is the value of n in 5^n=80n?

The value of n cannot be determined without further information. This equation has multiple solutions, and the value of n will depend on the specific context or problem being solved.

3. How do I solve for n in 5^n=80n?

To solve for n, you can use algebraic manipulation, logarithms, or graphing methods. The specific method used will depend on the context and the level of precision needed in the solution.

4. Can this equation be solved without using logarithms?

Yes, this equation can be solved without using logarithms. However, logarithms are often the most efficient and accurate way to solve this type of equation.

5. What are some common mistakes to avoid when solving 5^n=80n?

Some common mistakes to avoid when solving this equation include forgetting to take the logarithm of both sides, using incorrect properties of logarithms, and making calculation errors. It is important to double check your work and be careful with algebraic manipulations.

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