Question 886287: 2e^(x+3) = π^x
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 2*e^(x+3) = pi^x
divide both sides of this equation by 2 to get:
e^(x+3) = pi^x/2
take natural log of both sides of this equation to get:
ln(e^(x+3) = ln(pi^x/2)
since log(a^b) = b*log(a) and log(a*b) = log(a) + log(b), your equation becomes:
(x+3)*ln(e) = ln(pi^x) = ln(2)
since ln(e) = 1 and log(a^b) = b*log(a), your equation becomes:
(x+3) = x * ln(pi) - ln(2)
add ln(2) to both sides of this equation and subtract (x+3) from both sides of this equation to get:
ln(2) = x * ln(pi) - (x+3)
simplify this to get:
ln(2) = x*ln(pi) - x - 3
factor out the x to get:
ln(2) = x * (ln(pi) - 1) - 3
add 3 to both sides of the equation to get:
ln(2) + 3 = x * (ln(pi) - 1)
divide both sides of the equation by (ln(pi) - 1) to get:
(ln(2) + 3) / (ln(pi) - 1) = x
that's your solution.
to confirm the solution is correct, replace x in your original equation with the value of x in the solution to see if the equation holds true.
decimal equivalent of x = 25.51751602
after replacing x with 25.51751602, I got:
2*e^(x+3) = pi^x becomes:
4.853218453 * 10^12 = 4.853218453 * 10^12
this confirms the solution is correct.
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