My math skills are a bit rusty :(
I'd like to isolate the exponent "t" in this equation.
$$ 1.1 ^ t + 1.2 ^ t + 1.5 ^ t = 1,000,000,000 $$
So if I apply log on both sides, I'd have this:
$$ \log(1.1 ^ t + 1.2 ^ t + 1.5 ^ t) = \log(u) $$
So is $$\log(1.1 ^ t + 1.2 ^ t + 1.5 ^ t)$$ equals to $$\log(1.1 ^ t) + \log(1.2 ^ t) + \log(1.5 ^ t)$$???
Thank you!
My feeling is that it's not solvable, analytically
This list of logarithmic identities gives the identity $$\log\left(a_0 + a_1 + a_2\right) = \log a_0 + \log\left(1 + \frac{a_1}{a_0} + \frac{a_2}{a_0}\right).$$ Analytically, I think the best thing you'd be able to with that would be to write it as $\log(1 + x)$ and use taylor expansion, which would be very tedious, and would still not give you an exact answer.
My suggestion would be to use a numerical root-finding algorithm