How do I algebraically solve an equation that has exponential on one side and natural log on the other?

Question

The problem I'm stuck on is 1=ln(x-1)+e^(x-2). I solved this by graphing but I can't figure out how to isolate x algebraically since when I try to exponentiate or ln each side, the other operator remains.

Answer 1

Note if x = 2
1 = ln(2 - 1) + e^(2-2)
1 = ln 1 + e⁰
1 = 1

Answer 2

It looks like you are stuck with a transcendental equation, so your only approach may be a numerical one that gives approximate solutions.  I suggest plotting (which you already did), then obtaining graphically an initial guess for the root(s), and then running the initial root guess through an iterative procedure as in Newton's method.  Are you familiar with Newton's method, or with other numerical methods?

Answer 3

1=ln(x-1)+e^(x-2).  You already solved it in an excellent way (because this is a fairly oddball problem.)

 

1-ln(x-1) = e^(x-2).

 

ln [1-ln(x-1) ] = x-2.

 

Looks pretty hopeless, doesn't it!  Thanks to technology, you have solved the problem, empirically, and that's about all anyone can do, I believe.