Natural log of a variable times the variable

[ill]

so I've got this equation down to

x*ln(2/x) = 5

Can someone show/point me to a page where I can see how to solve this. I am under the impression it requires some type of ratio computation.

thanks!

 

[gb7nash]

It can't be solved for algebraically. You'll have to use software or an iterative technique. Wiki has a pretty good description of one technique called the bisection method:

http://en.wikipedia.org/wiki/Bisection_method

 

[olivermsun]

so I've got this equation down to

x*ln(2/x) = 5

Can someone show/point me to a page where I can see how to solve this. I am under the impression it requires some type of ratio computation.

thanks!

(hint: Try graphing the expression on the left.)

 

[uart]

so I've got this equation down to

x*ln(2/x) = 5

There are no real solutions to that equation. Are you interested in how to find the complex solutions?

 

[CellCoree]

I would approach this problem by first understanding the properties of natural logarithms and how they can be used to solve equations. The natural logarithm, ln(x), is the inverse of the exponential function, e^x. This means that ln(x) can be written as y = ln(x) if and only if x = e^y.

Now, looking at the equation x*ln(2/x) = 5, we can see that we have a product of two variables, x and ln(2/x). To solve for x, we need to isolate it on one side of the equation. We can do this by dividing both sides by ln(2/x):

x*ln(2/x)/ln(2/x) = 5/ln(2/x)

x = 5/ln(2/x)

Now, to solve for x, we need to find the value of ln(2/x). This requires understanding the properties of logarithms, specifically the power rule which states that ln(a/b) = ln(a) - ln(b). Applying this rule to ln(2/x), we get ln(2/x) = ln(2) - ln(x).

Substituting this into our equation for x, we get:

x = 5/(ln(2) - ln(x))

We can now solve for x by rearranging the equation:

x*ln(2) - x*ln(x) = 5

x*ln(2) = 5 + x*ln(x)

x*ln(2) - x*ln(x) = 5

x*(ln(2) - ln(x)) = 5

x = 5/(ln(2) - ln(x))

This is known as a transcendental equation, meaning it cannot be solved algebraically. However, it can be solved numerically using methods such as Newton's method or the bisection method. I would recommend consulting a math textbook or online resources for more information on these methods.