Find the number of solutions of the following equations: 1.`x^(-1/2)(log)_(0. 5)x=1, 2. `x^2-4x+3-(log)_(2)x=0`

Explanation

(i) We have , ` x^(-1//2) log_(0.5) x = 1`
`rArr log_(0.5) x = sqrtx`
To find the number of solutions, we must draw the graphs of `y=sqrtx and y = log_(0.5) x` and find the number of points of their intersection.
Graphs of functions are as shown in the following figure.

From the figure, we see that graphs intersect at only one point.
Hence, there is only one solution.
(ii) We have ` x^(2)-4x+3-log_(2) x = 0`
or ` x^(2) - 4x+3 = log_(2) x`
Let us draw the graphs of ` y = x^(2) - 4x + 3 and y = log_(2) x`.

From the figure, we see that graphs intersect at two points.
Hence, there are two solutions.