# a^x = b^x + c

• October 19th 2010, 04:14 AM
ayoyoayoyo
a^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.
• October 19th 2010, 06:44 AM
wonderboy1953
Quote:

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

for the life of me i cant do it.

To get you started, take the natural log of both sides of the equation.
• October 19th 2010, 06:45 AM
ayoyoayoyo
tried that but the constant term got in the way
• October 19th 2010, 06:48 AM
wonderboy1953
Quote:

Originally Posted by ayoyoayoyo
tried that but the constant term got in the way

Isn't it true that ln (ax) = ln(a) + ln(x)?
• October 19th 2010, 06:50 AM
ayoyoayoyo
i can not see how that pertains to the question at hand

we have ln(b^x+c)
• October 19th 2010, 07:00 AM
wonderboy1953
Quote:

Originally Posted by wonderboy1953
Isn't it true that ln (ax) = ln(a) + ln(x)?

I'll go further:

ln(a^x) = ln(b^x +c)= ln(b^x)ln(c), then

xln(a) = xln(b)ln(c); ln(c) = xln(a)/xln(b) = ln(a)/ln(b) = ln(a) - ln(b)
• October 19th 2010, 07:04 AM
Ackbeet
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?
• October 19th 2010, 07:58 AM
ayoyoayoyo
so no analytic solution?
• October 19th 2010, 08:01 AM
Ackbeet
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.
• October 19th 2010, 08:08 AM
ayoyoayoyo
how do you use newton's to find the solution? isnt newtons only used to find the roots of a f()?
• October 19th 2010, 08:11 AM
Ackbeet
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:

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

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

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.