What are the steps that I take to solve the following exponential equation:
e^x - 12^(e-x) - 1 = 0
I currently have gotten this far, but need assistance, as my method has gotten me the wrong answer:
e^x - 12^(e-x) - 1 = 0 Original exponential equation.
e^x - 12^(e-x) = 1 Add 1 to both sides of equation to isolate the exponential stuff.
-12^(e-x) = 1 - e^x Subtract e^x on both sides of equation to isolate 12^(e-x)
12^(e-x) = e^x - 1 Multiply both sides of the equation by -1.
log[12^(e-x)] = log(e^x - 1) Take common log of both sides, to start isolating x.
log(12) * (e-x) = log(e^x - 1) Property of Logarithms: Coefficients and Exponents. Simplify.
e - x = log(e^x - 1) - log(12) Subtract log(12) on both sides to isolate e - x.
e - x = log[(e^x - 1) / 12] Property of Logarithms: Subtraction & Quotients. Simplify.
-x = ln {log[(e^x - 1) / 12]} Take the ln of both sides of equation to isolate x.
x = -ln {log[(e^x - 1) / 12]} Multiply both sides of equation by -1 to find what x is.
The reason I know this is wrong, aside from the unnecessary problem of having to take ln of a log of a quotient, is because the website I must submit my answer to says the answer is a whole number. Any help would be appreciated, and if it is difficult to read anything, I sincerely apologize.
NOTE: If I divided both sides of the equation by log(12) on step 6, I would have ended up in the same place as I was in step 8. I was aware of this error, but if I came up with the same solution as the one if I did divide both sides of the equation by log(12), even with mathematics strict rules, it still works out in this situation.
