How to solve 10^[(10^(x+11)) +(0.5^(x+11))] =0

Question

anonymous · Answer

not possible i don't think a^(n) = 0 is possible

anonymous · Answer

Well, this is what I had before....log(log(x+11))=log^((1/2)^(x+11)).

anonymous · Answer

oh okay

anonymous · Answer

log^((1/2)^(x+11))??

anonymous · Answer

yeah

anonymous · Answer

i am assuming it's 

$$\log_{10} \log_{10} (x+11) = \log_{10} \frac{1}{2^{x+ 11}}$$

anonymous · Answer

okay yeah...

cristiann · Answer

The equation
log(x+11)=((1/2))^{x+11}
has a unique solution, which cannot be found exactly (or guessed ... :) )
The existence is proved by Darboux-type reasoning:
for x=-10: log1=0<(1/2)^1=1/2
for x=-1: log10=1>(1/2)^10
Unicity is proven by monotonicity ... :)

cristiann · Answer

The unique solution may be found numerically to be x=-9.5559
Darboux-type reasoning refers to:
f(x1)<0 and f(x2)>0 and f() continuous then there is at least one value x0 between x1 and x2 such that f(x0)=0

cristiann · Answer

:)

anonymous · Answer

wow cristiann, you're awesome at mathematics

cristiann · Answer

Thanks ... not really ... just older ... :)

cristiann · Answer

And I'm taking you all the fun of doing them ...:)

anonymous · Answer

REFER http://www.wolframalpha.com/input/?i=solve%7B10%5E%5B%2810%5E%28x%2B11%29%29+%2B%280.5%5E%28x%2B11%29%29%5D+%3D0%7D