SOLUTION: In the equation y=logbx, is it possible for x to have a negative value ? explain algebraically and graphically

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Question 252255: In the equation y=logbx, is it possible for x to have a negative value
? explain algebraically and graphically
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
In the equation y=logbx, is it possible for x to have a negative value
? explain algebraically and graphically


Algebraically:

y=log%28b%2Cx%29 means by definition b%5Ey=x

If y is positive, then x=b%5Ey is positive

If y is 0, then x=b%5Ey=b%5E0=1 which is also positive

If y is negative, then x=b%5E%28-y%29+=+1%2F%28b%5Ey%29 which is positive.

Therefore x can never be negative.

The graphs of 

red%28y=log%282%2Cx%29%29, green%28y=log%283%2Cx%29%29, and blue%28y=log%285%2Cx%29%29

are plotted below

graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Clog%282%2Cx%29%2Clog%283%2Cx%29%2Clog%285%2Cx%29%29 

And the graphs of 

red%28y=log%281%2F2%2Cx%29%29, green%28y=log%281%2F3%2Cx%29%29, and blue%28y=log%281%2F5%2Cx%29%29

are plotted below


 


So you see there is never any graph in the 3rd or 4th quadrants
where x is negative.

However in higher mathematics there are logarithms of negative
numbers but they are imaginary numbers.  If you have a TI-84
calculator 

PRESS MODE
scroll down to REAL
Press the right arrow key once
Press ENTER
Press 2nd
Press MODE
Press CLEAR
Press LOG
Press (-)
Press 1
Press )
(Should see log(-1) on the screen
Press ENTER
Read 1.364376354i

See? It's imaginary.  It has an "i" on the end. 
You won't study that unless you take higher math 
in college.  You teacher may not even know that.

Edwin