solution for exponential equation

Aug 2011
143
1
Hi,

Following is the problem today I encountered:

for what values of k that the equation 10 - 8(0.5)^x = k has (i) one solution and (ii) no solutions?

randomly I put values and checked; for many values of k there is one solution. For many values of k there is no solution.

For example I tried with 0, 1, 2, 3, etc.

It appears to be very crude and I am not able pin point rather, concisely or precisely say, this is the solution.

How to solve such type of problems?

guide me.

with regards,

Aranga
 
Dec 2014
153
117
USA
rewrite the equation as $10-k = 8(0.5)^x = 2^3 \cdot 2^{-x} = 2^{3-x}$

note that for all $x$, $2^{3-x} > 0$. If $k \ge 10$, there is no solution.

also note that the function $y=2^{3-x}$ is one to one; what does that say about the number of solutions for $x$ if $k < 10$ ?
 
Aug 2011
143
1
Thank you very much. I understood. Your reply is very useful.
Thanks.