ln(e) = ln(√(2)/x) -ln(e)

2 ln e = ln e^2

so
ln e^2 = ln (sqrt 2 / x)

e^2 = sqrt 2/x

x = sqrt 2 / e^2

ln e = 1 , so your equation becomes

1 = ln(√(2/x) - 1
2 = ln √(2/x)
√(2/x) = e^2
square both sides
2/x = e^4
x = 2/e^4

notice how I read your expression
If you meant
ln e = ln (√2 /x) - ln 2
2 = ln (√2/x)
√2/x = e^2
x = √2/e^2

To solve the equation ln(e) = ln(√(2)/x) - ln(e), we can simplify it step by step.

1. Simplify the natural logarithm of e:
ln(e) = 1

2. Simplify the natural logarithm of (√(2)/x):
ln(√(2)/x) = ln(√2) - ln(x)

3. Rewrite the equation after simplification:
1 = ln(√2) - ln(x) - 1

4. Simplify further:
1 = ln(√2) - ln(x) - 1
1 + 1 = ln(√2) - ln(x)
2 = ln(√2) - ln(x)

Now, if you want to calculate the value of x, you need to isolate ln(x) on one side of the equation. Let's continue:

5. Add ln(x) to both sides of the equation:
2 + ln(x) = ln(√2)

6. Since ln(x) is being added to 2, we can apply the property that ln(a) + ln(b) = ln(ab):
ln(x * e^2) = ln(√2)

7. By equating the arguments of the natural logarithms, we can conclude that:
x * e^2 = √2

8. Finally, to find the value of x, divide both sides of the equation by e^2:
x = √2 / e^2

Therefore, the solution to the equation is x = √2 / e^2.