Solving for variable N when it is used as a base and exponent

In summary, the conversation discusses solving the equation (n/2)^n = c for the variable n, using logarithm properties. The attempt at a solution involves using logs of base 2 and getting stuck when trying to manipulate the logarithms further. The conversation also touches on the use of natural logs when dealing with variables.
  • #1
wjang
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Homework Statement



I would like to solve the following equation for the variable "n"
The variable c represents any integer constant.

(n/2)^n = c

The goal is to get this in some form of n = ?

Homework Equations



The logarithm properties.

The Attempt at a Solution



I've tried using logs but I get stuck because there seems to be no way to somehow combine n as a base, and n as an exponent. Is this something that requires higher level math?

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Let all logs be of base 2

(n/2)^2 = c
log(n/2)^n = log(c)
n*log(n/2) = log(c)
n(log(n)-log(2)) = log(c)
n(log(n)-1) = log(c)

Here I get stuck because any further logarithm manipulation only goes in circles.

Can anyone teach or point to me what kind of math I need to know to solve this please?
 
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  • #2
Why didn't you move -log(2) over to the other side? Also, I like natural logs better when you're dealing with variables because you don't have to worry about the base.
 

FAQ: Solving for variable N when it is used as a base and exponent

1. How do you solve for variable N when it is used as a base and exponent?

To solve for variable N when it is used as a base and exponent, you can use logarithms. Take the logarithm of both sides of the equation, and then use logarithm rules to simplify the expression.

2. Can you give an example of solving for variable N when it is used as a base and exponent?

Sure! For example, if we have the equation N^2 = 16, we can take the logarithm of both sides to get log(N^2) = log(16). Then, we can use the power rule of logarithms to simplify the expression to 2log(N) = log(16). Finally, we can use the logarithm property log(a^b) = b*log(a) to solve for N, giving us N = 4.

3. Are there any other methods for solving for variable N when it is used as a base and exponent?

Yes, there are other methods such as using the inverse function of exponentials, which is the logarithmic function. You can also use algebraic manipulation to isolate N on one side of the equation. However, logarithms are the most commonly used method for solving these types of equations.

4. Can you solve for variable N if it is used as both a base and exponent in a more complex equation?

Yes, the method of taking the logarithm of both sides and using logarithm rules can be applied to more complex equations as well. It may require some algebraic manipulation, but the same principles still apply.

5. Are there any special cases to consider when solving for variable N when it is used as a base and exponent?

Yes, there are a few special cases to consider. If the base is negative, you will need to use complex numbers and the natural logarithm. Also, if the exponent is a fraction, you will need to use logarithms with different bases and the change of base formula. And finally, if the base is 1, the solution will always be 1 regardless of the exponent.

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