Problem 8. What value of x satisfies the equation 6 · e2x = 4x? (A) 2ln6 ln4 (B) ln4 2ln6 (C) ln4−2 ln6 (D) ln6 ln4−2 (E) 0

Answer 1

ln 6 + ln e^2x = ln 4 + ln x

ln 6 + 2x = ln 4 +ln x

ln 6 - ln 4 - ln x = -2x

[-ln (6 / 4x)] / 2 = x

Answer 2

6e^2x =4x has no real solution

maybe answer e) was intended to be the null set, not zero

a 0 with a slash through it

then it would be the correct answer

c) is temptingly "close"

as you might try to take natural logs of both sides

and get

ln6 + 2x = ln4 +lnx

and

x = (-ln6+ln4+lnx)/2

but that's x solved in terms of x

and it's not a solution

it's just rewriting the equation

try to isolate x and there is no real solution

plus c) left got the 2 wrong and left out x in the purported "solution"

that "6*e^2x" part looks suspicious though. Maybe it meant something other than (6)(e^2x), which would change the problem

or

another possibility

4^x instead of 4x

6e^2x = 4^x

ln6 + 2x = ln(4^x) =xln4

2x-xln4 = -ln6

x(2-ln4) = -ln6

x = -ln6/(2-ln4) = -.5ln6 + ln6/ln4

x =-(1/2)ln6 +log₄6

but none of the choices fit that answer either