Question

Find the number of real solution of the equation \[{\log _{0.5}}x = |x|\]
A) 1
B) 2
C) 0
D) None of these

Answer 1

Hint: Try to solve the question graphically. First draw the graph of \[y = {\log _{0.5}}x\] then draw the graph of \[y = |x|\] and observe how many times the graph meets.
That will give us the correct answer.

Complete Step by Step Solution:
Considering the given equation
\[{\log _{0.5}}x = |x|\]
It can be stated that the domain of log function is only at the positive side of the x-axis.
Therefore the graph will be like the figure below

Now let's try to draw the graph of \[y = |x|\] . It will be a straight line containing the points \[(0,0),(-1,1),(2,2).....\]

Now if we merge both the diagrams we will observe that it cuts only in one point

From the figure it is clear that both the graphs cut only once on the real axis
Thus, there will be only one real solution of the equation \[{\log _{0.5}}x = |x|\]
Therefore option A is correct.

Note: There is no mod sign in x for the equation \[y = {\log _{0.5}}x\] that's why we only took one side of the logarithmic graph(+ve side) and due to mod sign in the equation \[y = |x|\] we took both sides of the graph. Also note that the base of the log is less than 1 that's why the graph is looking inverse then that of the logs with bases greater than or equal to 1.