Can anyone help me solve this LN problem?

Question

How do I solve the following:

2ln(x^(1/2)) - ln(1-x) = 2

I am totally lost.

Answer 1

x/(1-x) = e^2

x = (1-x)e^2

(1+e^2)x = e^2

x = e^2/(1+e^2)

= 0.88

Answer 2

You just have to know a few "tricks" to use with ln.

First trick, and my personal favorite, is ln (x^(y)) = y*ln(x)

That means that you can "move" the exponent of the function in front of the ln and mutliply it. So for you:

2 ln(x^(1/2)) = (2 * 1/2) ln(x) = ln (x)

The second trick is ln (x) - ln (y) = ln (x/y)

That means that you can divide the first function by the second function. For you:

ln (x) - ln(x-1) = ln (x/(x-1))

The last trick is when you raise an ln function to the power of "e", that cancels out the ln. For instance, e^(ln(x)) = x.

For you: e^(ln(x/(x-1)) = e^2

x/(x-1) = e^2

And you can solve it from there using fun old algebra

Or, just read the guy above me, haha.