# solution for exponential equation

#### arangu1508

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

#### Cervesa

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 $k < 10$ ?