solve e^x = log x?

Question

sorry i meant ln x
solve e^x= ln x

Answer 1

graph them, see if they intersect (they don't). So there is no solution. (This is assuming that you are looking for a real number solution. I forgot how to find the complex solution.)

Answer 2

e^x = log x ↔ 10^(e^x) = 10^(log x) = x.

If x ≥ 1, then x < e^x < 10^(e^x). Thus, no solution exists for x ≥ 1.

Go back to your original equation.

e^x = log x ↔ x = ln(e^x) = ln(log x).

If x < 1, then log x < x → ln(log x) < ln x < x. Thus, no solution exists for x < 1.

Combining these, we see that no solutions exists. Now try to find an easier method that works in one step.