#### To determine

**To find:** We have to solve equation ** e2x-ex-6=0** for x.

#### Answer

The solution of the equation is x=ln3.

#### Explanation

**Calculation:**

We have to solve equation** e2x-ex-6=0**

**Step 1:** First we write the equation in quadratic form and then factorize it

(ex)2-3ex+2ex-6=0, or

ex(ex- 3)+2 ex-3=0,

(ex-3)(ex+ 2)=0

The only way a product of two factor is zero is when one or both of the factor is zero. Therefor

Either ex-3=0 or ex+2=0, that is either ex=3 or ex=-2.

Since, we know that exponential function takes only positive values for all values of x, thus ex≠-2.

**Step 2: ** Thus ex=3. Taking natural logarithmic on both sides of the equation, we have

lnex=ln3.

**Step 3:** Since we know that lnex=x, thus, we have

x=ln3.

**Conclusion:**

The solution of the equation is x=ln3.