How to solve exponential function of form

peggyleuwpodg

peggyleuwpodg

Answered question

2022-03-30

How to solve exponential function of form abx+x=c?
For instance, how would I solve: 3x+x=85 ?

Answer & Explanation

Mikaela Winters

Mikaela Winters

Beginner2022-03-31Added 14 answers

abx+x=c
abx=cx
bx=cxa
exlnb=cxa
1=cxaexlnb
eclnb=cxaexlnbeclnb
aeclnb=(cx)e(cx)lnb
lnb.a.eclnb=lnb(cx)e(cx)lnb
bclnb.a=lnb(cx)e(cx)lnb
u=lnb(cx)
ueu=bclnb.a
u=W(bclnb.a)
where W(x) is Lambert W function
u=lnb(cx)=W(bclnb.a)
x=cW(abclnb)lnb
It is general solution of abx+x=c
For your example:
3x+x=85
x=85W(385ln85)ln85

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