Exponential equation

[sadovnichek]

Hi everyone.

I want to solve the next equation:
$$2^{n} = n + 2$$
I don't know how to get complete solution. First, I try to solve with graphical method and find one root - it's n = 2. The second one placed between (-2, -1.5). Then I began an analytical solution and stuck again:
$$\log_{2}(2^n) = \log_{2}{(n + 2)}$$$$n = \log_{2}{(n + 2)}$$???
Any ideas?

 

[BigBeachBanana]

Hi everyone.

I want to solve the next equation:
$$2^{n} = n + 2$$
I don't know how to get complete solution. First, I try to solve with graphical method and find one root - it's n = 2. The second one placed between (-2, -1.5). Then I began an analytical solution and stuck again:
$$\log_{2}(2^n) = \log_{2}{(n + 2)}$$$$n = \log_{2}{(n + 2)}$$???
Any ideas?

What's n's domain? Real, Integer, Complex...?

 

[Steven G]

This problem can't be solved algebraically.

 

[sadovnichek]

What's n's domain? Real, Integer, Complex...?

Real

This problem can't be solved algebraically.

Why?

 

[Steven G]

You have n in the base and n in the exponent in the same equation. Sometimes the answers can be seen my inspection, but no algebraic manipulation will work.

 

[BigBeachBanana]

WolframAlpha provides the general solution via Lambert's W function. $W_m(k)$.


The solutions are $n\approx-1.69009 \land n=2$, which can also be seen by graphing.

 

[Deleted member 4993]

Why?

Why can't you run faster than 70 miles/hr?

 

[Deleted member 4993]

n=log₂(n+2)

The solution can be approximated using numerical methods (Newton-Raphson) or by graphical method.

 

[Steven G]

Why can't you run faster than 70 miles/hr?

What does that have to do with the OP's question?

 

[Deleted member 4993]

What does that have to do with the OP's question?

When you had responded (response # 3) - the OP responded

Why?

and I provided the answer with:

Why can't you run faster than 70 miles/hr?

Why is responded with another "why".

Science cannot help me answer "why" - I can only try to answer "how".