Solve the following exponential equation. Express irrational solutions in exact form.

Question

Solve the following exponential equation. Express irrational solutions in exact form.

5^2x + 5^x - 12

Answer 1

5^2x = (5^2)(5^2). Let z = 5^2 then

5^2x + 5^x - 12 =

z^2 + z -12. set it equal to zero, factor and solve for z

(z+4)(z-3) = 0

z = 3 and -4

5^x =3 and 5^x= -4

take log to the base 5

log(5^x) = log 3

x = log of 3 to the base 5

log(5^x) = log (-4)

x = log of -4 to the base 5

but 5 to any power will never be negative, so ignore this "solution"

only answer is x = log of 3 to the base 5 = whatever exponent you need to put over 5 to get 3. It's somewhere between 0 and 1, as 5^0 = 1 and 5^1 = 5. It's about 0.6828

But for an "exact" solution, then it's just log of 3 to the base 5. log₅3

which is about 7/10 or .7

While you can ignore the negative log as having no real solution, Euler used his identity to find imaginary solutions.

e^ix = cosx + isinx

let x = pi

e^ipi = cospi + isinpi = -1

take natural logs of both sides to get

ipi = ln(-1)

ln(-1)=ipi

so the log of a negative number does have a solution, but it's not real, "just' imaginary

For completeness you could calculate the 2nd solution of this equation. But it's not real.