Find intersection of linear and logarithmic lines I have

Ormezzani6cuu

Ormezzani6cuu

Answered question

2022-04-12

Find intersection of linear and logarithmic lines
I have equations for two lines, one of which is linear and the other is logarithmic, ie:
y=m1x+c1
y=m2ln(x)+c2
..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:
m1x+c1=m2ln(x)+c2
..for x, but apart from shuffling the constants around I'm not sure how to do this
Is there a general solution to this problem?
Thanks

Answer & Explanation

muthe2ulj

muthe2ulj

Beginner2022-04-13Added 10 answers

The general solution involves the Lambert W function, whose defining equation is z=W(z)eW(z) for complex numbers z. If either m1 or m2 is zero in the given problem, then the solution is elementary, so suppose m1,m20. Then
m1x+c1=m2lnx+c2
can be rewritten as
m1m2ec1c2m2=m1xm2em1xm2,
which has the solution
m1xm2=W(m1m2ec1c2m2),
or
x=m2m1W(m1m2ec1c2m2).
For instance, if m1=1,m2=1, and c1=c2=0 this gives x=W(1)=Ω=0.56714329 (the Omega constant), which is correct, since Ω=lnΩ

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