Which of the following logarithmic equations has no solution?

Question

A) log4(x + 2) = log4(6 - x)

B) log4(x + 2) = 1

C) log4(x+2) = log4(2x+6)

D) log4(-x) = 1

 

*** The 4's should be bases

Answer 1

If we were to solve for x by applying the rules and properties of logs, then plug in those x values into the original equation, then the equation should be satisfied.  Also, note that the argument of the log after plugging in the x values must be positive.

 

Lets look at choice C.  If we equate the log's arguments, we attain

 

x + 2 = 2x + 6

 

 

Solving for x gives us

 

-4 = x

 

 

Plugging in this value of x into the original equation yields

 

log₄(-4 + 2) = log₄(2(-4) + 6)

 

log₄(-2) = log₄(-2)

 

 

Since the argument of the log is negative, there is no solution.  This of it this way.  4 raised to the what power results in -2?  There is no power.  If a positive number is raised to a number greater than or equal to 0, then the result is positive.  If a positive base is raised to a negative power, then the result is a number between 0 and 1.