So, I was asked recently if I could help solve a simple exponential equation. And, while I was able to get an answer, I was only able to do so by inspection. I'm not sure how to go about solving a more general equation without using numerical methods.
Anyway, here's the equation:
SQRT( (3 + SQRT(8))^x ) + SQRT( (3 - SQRT(8))^x ) = 6
When I took a look at this, I was able to solve it by inspection readily...
The equation can be rewritten as:
(3 + SQRT(8))^(x/2) + (3 - SQRT(8))^(x/2) = 6
Clearly, if we had (3 + SQRT(8)) + (3 - SQRT(8)), we would get 6. So, we just want the exponents to be equal to 1, and thus, x = 2.
That's all fine and dandy, but that doesn't really sit well with me.
What if we had the following instead:
SQRT( (3 + SQRT(8))^x ) + SQRT( (3 - SQRT(8))^x ) = C
Now what? How would we be able to solve this in terms of C? Now, is there some sort of identity or trick that we can take advantage of to help us solve this... maybe something related to the conjugate?
I thought about this more, but I got nowhere. In terms of a quick and dirty approximation, we could simply drop the 2nd term knowing that as C tends to infinity, the (3 - SQRT(8))^(x/2) would approach zero.
So, for large C we could get an accurate approximation by simply solving the simpler equation:
(3 - SQRT(8))^(x/2) = C
And, x = (2 ln C) / (ln (3 + SQRT(8)). The value of C doesn't even have to get all that big for that approximation to be accurate. At C = 6, we have 1.65% error. At C = 34, we are down to a tiny 0.0246% error. And, by the time C reaches 198, we have a near negligible 0.000482% error.
Okay, but it's still not that useful. And, for small values of C, we are certainly going to be way off the mark.
Continuing with all of this... what do you do with another simple exponential equation like:
2^x + 3^x = C
I could get nowhere in trying to break this down. Is the only way to get a value for x is by solving for it numerically using known methods?
I've either forgotten a lot of simple math or there's just no clean way to solve exponential equations that contain different bases. Anyway, I'm done thinking about this for now. So, does anyone want to help me out with this?