- WE WANT TO FIND THE EQUATION OF THE VERTICAL ASYMPTOTE
TO EACH LOG FUNCTION.
OUR FIRST FUNCTION IS F OF X = LOG OF THE QUANTITY X - 4.
NOTICE HOW THERE'S NO BASE GIVEN.
SO WE KNOW THIS IS BASE 10 OR A COMMON LOG.
SO WE COULD REWRITE THIS AS Y = LOG BASE 10
OF THE QUANTITY X - 4.
WE SHOULD KNOW BY NOW THAT THIS IS EQUIVALENT
TO THE EXPONENTIAL EQUATION 10
RAISED TO THE POWER OF Y = THE QUANTITY X -4.
SO WE KNOW X CAN BE ANY VALUE THAT WOULD BE EQUAL TO 10
RAISED TO THE POWER OF Y,
BUT 10 RAISED TO THE POWER OF Y
IS ALWAYS GOING TO BE POSITIVE.
THAT'S THE REASON WHY TO FIND THE DOMAIN,
WE SOLVE THE INEQUALITY X - 4 > 0,
AND THEN TO FIND THE EQUATION OF THE VERTICAL ASYMPTOTE,
WE NEED TO SOLVE THE EQUATION X -4 = 0.
SO IF WE SOLVE THIS INEQUALITY FOR X,
WE WOULD ADD 4 TO BOTH SIDES.
THE DOMAIN IS X > 4,
AND THEN IF WE SOLVE THIS EQUATION FOR 4,
WE'D HAVE X = 4,
WHICH WOULD BE THE EQUATION OF THE VERTICAL ASYMPTOTE.
SO NOTICE THAT THE X VALUE CAN GET CLOSER AND CLOSER
TO POSITIVE 4.
IT JUST CAN'T = +4,
BECAUSE IF IT DID, WE'D HAVE 10 TO THE POWER OF Y = 0,
WHICH CANNOT OCCUR.
NOW, WE'LL TAKE A LOOK AT THIS GRAPHICALLY AS WELL
IN JUST A MOMENT.
BUT LET'S TAKE A LOOK AT OUR 2nd EXPONENTIAL FUNCTION.
WE HAVE F OF X = NATURAL LOG OF THE QUANTITY X + 1.
REMEMBER, NATURAL LOG IS LOG BASE E.
SO WE COULD WRITE THIS AS Y = LOG BASE E
OF THE QUANTITY X + 1,
AND THEN IN EXPONENTIAL FORM,
WE'D HAVE E RAISED TO THE POWER OF Y
= THE QUANTITY X + 1.
SO IN THIS FORM IT'S EASIER TO SEE THE DOMAIN
WOULD HAVE TO OCCUR WHEN X + 1 > 0,
AND THEREFORE, THE EQUATION OF THE VERTICAL ASYMPTOTE
WOULD OCCUR WHERE X + 1 = 0.
AGAIN, FOR THE DOMAIN
WE SOLVED THE INEQUALITY X + 1 > 0,
AND THEN FOR THE VERTICAL ASYMPTOTE
WE SOLVED THE EQUATION X + 1 = 0.
SO SUBTRACT 1 ON BOTH SIDES.
WE HAVE X > -1 FOR THE DOMAIN,
AND THEN WE SUBTRACT 1 ON BOTH SIDES OF THE EQUATION.
WE HAVE X = -1 FOR THE EQUATION
OF THE VERTICAL ASYMPTOTE.
LET'S VERIFY THIS GRAPHICALLY AS WELL.
AGAIN, OUR FIRST LOG FUNCTION WAS COMMON LOG OR LOG BASE 10.
SO IN EXPONENTIAL FORM WE WOULD HAVE 10
TO THE POWER OF Y = QUANTITY X - 4.
TO GRAPH THIS BY HAND, WE'D SELECT VALUES OF Y
AND THEN SOLVE FOR X,
WHICH WOULD PRODUCE THIS BLUE GRAPH HERE.
NOTICE HOW THE GRAPH APPROACHES +4
BUT NEVER REACHES IT,
AND THAT'S WHY WE HAVE A VERTICAL ASYMPTOTE OF X = +4.
JUST KEEP IN MIND THE GRAPH NEVER DOES TOUCH
THIS VERTICAL LINE, X = 4,
AND THAT'S THE REASON WHY OUR DOMAIN IS X > +4.
AND THEN FOR OUR 2nd LOG FUNCTION,
THE NATURAL LOG FUNCTION, IN EXPONENTIAL FORM
WE WOULD HAVE E RAISED TO THE POWER OF Y
= THE QUANTITY X + 1.
WE TAKE A LOOK AT THE GRAPH OF THIS FUNCTION.
HERE'S OUR VERTICAL ASYMPTOTE OF X = -1.
THE GRAPH APPROACHES THIS LINE BUT NEVER TOUCHES IT,
AND OUR DOMAIN IS X > -1.
OKAY, HOPE THE EXPLANATIONS AS WELL AS THE GRAPHS
HELP YOU BETTER UNDERSTAND THE DOMAIN
AS WELL AS THE EQUATION OF THE VERTICAL ASYMPTOTES.