Solve the equation exact and rounded answers e3x 3e2x 1Sol

Solve the equation, exact and rounded answers e^3x = 3e^2x + 1

e^3x = 3e^{2x +1}

==> e^3x = 3e^2x e¹ since a^m+n = a^m aⁿ

==> e^3x/e^2x = 3e

==> e^{3x - 2x} = 3e since a^m/aⁿ = a^m-n

==> e^x = 3e

Apply natural logarithms on both sides

==> ln e^x = ln (3e)

==> x lne = ln3 + lne since ln a^b = b lna , ln(ab) = lna + lnb

==> x(1) = ln3 + 1 since log_a a = 1

==> x = 1 + ln3

==> x = 2.0986

Hence solution is x = 2.0986