# Problem solving equation of type x=a*b^x [closed]

I have the equation x = a*b^x and want to solve it for x. But every online solver I tried says that it is not possible.

But when I choose a==8 and b==0.5 there is a solution for x==2

Is it not possible to solve the equation formally?

-

## closed as off topic by Igor Rivin, Steven Landsburg, Emil Jeřábek, Bill Johnson, Todd Trimble♦Dec 6 '12 at 15:43

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.

The eager closers have done their (dirty) work here. If you want more information about this use of the Lambert W function, try asking at one of the other locations found in the FAQ. For example math.stackexchange.com –  Gerald Edgar Dec 6 '12 at 16:05
Thanks for your comment! –  Rick Dec 6 '12 at 21:14

There are no elementary formulas for such equations. Using the Lambert W function, the solution is $$x = -\frac{W(-a \log b)}{\log b}$$

-