maths logarithm help please...

Question

the number  of real solutions of the equation log_0.5 x = | x |  ?

 

•  1
•  2
•  0
•  none of these 

Answer 1

Suppose x is real.  Then, x>0, because the logarithm of zero or a negative number does not exist.

 

|x| becomes x.

 

The equation becomes

 

(1/2)^x=x=1/(2^x)

 

x2^x=1

 

y=x2^x is positive, has a positive slope and is 0 at x=0.  Therefore it crosses the line y=1 once.

 

If x is real, there is one real solution.

 

Suppose x is complex.

 

x=Re^iθ 

 

|x|=R

 

(1/2)^|x|=x

 

1/2^R=Re^iθ 

 

R2^Re^iθ =1

 

eiθ=1/(R2^R)

 

The range of y=1/(R2^R) is y>0, and it is real.

 

e^iθ must be real.

 

e^iθ =cos θ + i sin θ

 

Therefore, θ must be 0 ±n2pi radians, in which case e^iθ = 1.

 

So, R has the value found when x is real, and there are an infinite number of angles that imply an infinite number of solutions.  However, there is only one solution in the range R>0 and -pi<=θ<=pi radians.

 

The answer you want is "1".