how does one solve an equation of the type a^x = b^x + c??

for the life of me i cant do it.

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- Oct 19th 2010, 04:14 AMayoyoayoyoa^x = b^x + c
how does one solve an equation of the type a^x = b^x + c??

for the life of me i cant do it. - Oct 19th 2010, 06:44 AMwonderboy1953
- Oct 19th 2010, 06:45 AMayoyoayoyo
tried that but the constant term got in the way

- Oct 19th 2010, 06:48 AMwonderboy1953
- Oct 19th 2010, 06:50 AMayoyoayoyo
i can not see how that pertains to the question at hand

we have ln(b^x+c) - Oct 19th 2010, 07:00 AMwonderboy1953
- Oct 19th 2010, 07:04 AMAckbeet
The logarithm of a sum is not something you can simplify. The identity only works the other way.

With this problem, I'd probably go for a numerical solution using Newton-Raphson. Of course, this assumes you know a, b, and c, and that a and b are both positive. Is that the case? - Oct 19th 2010, 07:58 AMayoyoayoyo
so no analytic solution?

- Oct 19th 2010, 08:01 AMAckbeet
None of which I am aware. Both Mathematica and WolframAlpha fail to give an analytical solution. I should point out that not every combination of a, b, and c will admit a solution, even if all of them are positive.

- Oct 19th 2010, 08:08 AMayoyoayoyo
how do you use newton's to find the solution? isnt newtons only used to find the roots of a f()?

- Oct 19th 2010, 08:11 AMAckbeet
Right, but you can always convert an equation-solving problem into a root-finding problem by throwing everything on to one side of the equation thus:

$\displaystyle f(x)=a^{x}-b^{x}-c=0.$

Then you use Newton-Raphson on $\displaystyle f(x).$

If you have convergence problems, and your solutions are oscillating wildly, then you probably don't have a solution for that particular combination of a, b, and c. That's a little warning sign you can look for.