Question

How do you solve `(2lnx) + 1 = ln(2x)`?

Answer 1

`x = 2/e`

Explanation:

We will use the following:

• `ln(a^x) = xln(a)`
• `ln(a) - ln(b) = ln(a/b)`
• `e^ln(a) = a`

---

`2ln(x) + 1 = ln(2x)`

`=> ln(x^2) + 1 = ln(2x)` (by the first property above)

`=> ln(x^2) - ln(2x) = -1`

`=> ln(x^2/(2x)) = -1` (by the second property above)

`=> ln(x/2) = -1`

`=> e^ln(x/2) = e^(-1)`

`=> x/2 = 1/e` (by the third property above)

`:. x = 2/e`