Solve the exponential equation. express solutions in exact form only. `e^(2x)-10e^x/13 =3`

Expert Answers
txmedteach eNotes educator| Certified Educator

To take care of this question, you need to first solve this as a quadratic equation. Let's let `u = e^x`. This gives us the following equation:

`u^2 - 10/13 u = 3`

Now, we subtract 3 from both sides:

`u^2 - 10/13 u - 3 = 0`

We have a quadratic equation, that is, unfortunately, not easily factored. However, we can easily find the roots using the quadratic formula:

`u = (10/13 +- sqrt((10/13)^2 - 4*1*-3))/2`

Simplifying the square root:

`u = (10/13 +- sqrt(100/169 + 12))/2`

Now, we get the 12 term over a denominator of 169, and we add 100:

`u = (10/13 +-sqrt(2128/169))/2`

We can simplify the square root slightly:

`u = (10/13 +- 4/13 sqrt(133))/2`

Finally, we can divide both parts by 2 to get the following final result for u:

`u = 5/13 +- 2/13sqrt(133)`

Now, remember what `u` is supposed to be: `e^x`. Notice that if we take the solution where we subtract the radical term that we get a negative number. Therefore, our solution is found only with the following possibility:

`u = 5/13 + 2/13 sqrt(133)`

Now, we can solve by substituting `e^x` for `u`:

`e^x = 5/13 + 2/13 sqrt(133)`

Our final step is to take the natural log (the inverse of `e^x`):

`x = ln(5/13 + 2/13 sqrt(133))`

There is the final answer in exact form. I hope that helps!