Question 886287
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.