How do you solve 3^x= 3x+53x=3x+5?

1 Answer
Apr 3, 2018

See below.

Explanation:

Making y = 3x+5y=3x+5 we have

3^((y-5)/3) = y3y53=y or

3^(y/3)3^(-5/3) = y3y3353=y

now making 3^(1/3) = e^lambda313=eλ we have

e^(lambda y) 3^(-5/3) = yeλy353=y or

3^(-5/3)=y e^(-lambda y)353=yeλy or

(-lambda)3^(-5/3) = (-lambda y) e^(-lambda y)(λ)353=(λy)eλy

At this point we use the Lambert function

https://en.wikipedia.org/wiki/Lambert_W_function

Y = X e^X hArr X = W(Y)Y=XeXX=W(Y) and then

-lambda y = W((-lambda)3^(-5/3))λy=W((λ)353) or

y = -1/lambda W((-lambda)3^(-5/3)) = 3x-5y=1λW((λ)353)=3x5 and finally

x = (-5 Log 3 - 3 W(-(Log 3/(9 xx 3^(2/3)))))/(3 Log 3)x=5log33W((log39×323))3log3

We have then two solutions

x =-1.60981x=1.60981 and x = 2.24048x=2.24048

Attached a plot showing the intersections of y = 3^xy=3x and y = 3x+5y=3x+5

enter image source here