Solve the exponential equation. Express your solutions in exact form only. Please show all of your work.
(e^(2x) – 10e^x) / 13 = 3
To solve the exponential equation (e^(2x) – 10e^x) / 13 = 3, we can start by multiplying both sides of the equation by 13 to eliminate the fraction:
13 * ((e^(2x) – 10e^x) / 13) = 3 * 13
This simplifies to:
e^(2x) – 10e^x = 39
Now, let's make a substitution to simplify the equation. Let's introduce a new variable, u = e^x. This means that e^(2x) = u^2, and 10e^x = 10u.
Substituting these values into the equation, we get:
u^2 - 10u = 39
Rearranging the equation, we have:
u^2 - 10u - 39 = 0
Now, we can solve this quadratic equation for u by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
u = (-(-10) ± sqrt((-10)^2 - 4*1*(-39))) / (2*1)
This simplifies to:
u = (10 ± sqrt(100 + 156)) / 2
u = (10 ± sqrt(256)) / 2
u = (10 ± 16) / 2
This gives us two possible values for u:
u = (10 + 16) / 2 = 13
u = (10 - 16) / 2 = -3
We now substitute back u = e^x into the equation to find the values of x:
For u = 13, we have:
e^x = 13
Taking the natural logarithm (ln) of both sides, we get:
x = ln(13)
For u = -3, we have:
e^x = -3
However, since the exponential function is always positive, there are no real solutions for this case.
Therefore, the solution to the exponential equation (e^(2x) – 10e^x) / 13 = 3, expressed in exact form, is:
x = ln(13)