Ex:  Find the Domain of Logarithmic Functions
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Ex: Find the Domain of Logarithmic Functions


– WE WANT TO FIND THE DOMAIN
OF THE GIVEN LOG FUNCTIONS AND THEN STATE THE DOMAIN USING INEQUALITIES
AND INTERVAL NOTATION. LOOKING AT OUR FIRST FUNCTION, WE HAVE Y EQUALS LOG BASE 2
OF THE QUANTITY 2X MINUS 4. THE QUANTITY 2X MINUS 4
IS THE NUMBER PART OF THE LOG THAT’S OBTAINED BY RAISING 2
TO THE POWER OF Y. REMEMBER, IN EXPONENTIAL FORM, 2 IS THE BASE,
Y IS THE EXPONENT AND THE NUMBER WOULD BE
THE QUANTITY 2X MINUS 4. SO IF WE THINK ABOUT
ALL THE POSSIBLE VALUES WE GET WHEN RAISING 2 TO A POWER, WELL, THE RESULT IS ALWAYS
GOING TO BE GREATER THAN ZERO. THAT’S THE REASON
WHY TO FIND THE DOMAIN WE HAVE TO SOLVE
THE INEQUALITY 2X MINUS 4 IS GREATER THAN ZERO. SO THE NUMBER PART
OF THE LOGARITHM IS ALWAYS GOING TO BE GREATER
THAN ZERO. SO NOW WE’LL ADD 4
TO BOTH SIDES. THAT WOULD GIVE US 2X
IS GREATER THAN 4. DIVIDE BOTH SIDES BY 2. SO WE HAVE X IS GREATER
THAN 2. SO OUR DOMAIN WOULD BE X
GREATER THAN 2 USING INEQUALITIES. WE WANT TO EXPRESS THIS
USING INTERVAL NOTATION. IT MIGHT BE HELPFUL
TO SKETCH A GRAPH. SO IF THIS IS 2,
WE HAVE AN OPEN POINT ON 2, ARROW TO THE RIGHT
APPROACHING POSITIVE INFINITY. THEREFORE, USING
INTERVAL NOTATION WE HAVE THE INTERVAL
FROM 2 TO INFINITY AND IT’S OPEN ON 2
BECAUSE IT DOES NOT INCLUDE 2. OKAY. LOOKING AT OUR
SECOND EXAMPLE. NOTICE HOW THERE’S NO BASE
LISTED ON THIS LOG. THEREFORE WE KNOW IT’S
COMMON LOG OR LOG BASE 10. AND IF WE FOCUS
ON THE NUMBER PART OF THE LOG THE QUANTITY X SQUARED
MINUS 3X WOULD BE OBTAINED BY RAISING 10
TO THE POWER OF Y WHICH SHOULD ALWAYS BE GREATER
THAN ZERO. THEREFORE, TO FIND THE DOMAIN
OF THIS FUNCTION, WE NEED TO SOLVE THE
INEQUALITY X SQUARED MINUS 3X GREATER THAN ZERO. THE SOLUTIONS TO THIS WILL BE
THE DOMAIN OF OUR FUNCTION. TO SOLVE A QUADRATIC
INEQUALITY, WE FIRST FIND THE SOLUTIONS
AS IF IT WAS AN EQUATION. SO THIS DOES FACTOR SO WE WANT
TO FIRST SOLVE THE EQUATION X TIMES A QUANTITY X MINUS 3
IS EQUAL TO ZERO. WELL, THIS WOULD BE ZERO WHEN X EQUALS ZERO
OR WHEN X EQUALS 3. SO THESE VALUES OF X WHERE
IT WOULD BE EQUAL TO ZERO, WE WANT TO FIND THE X VALUES
WHERE IT’S GREATER THAN ZERO. SO WHAT WE’LL DO NOW
IS SKETCH A NUMBER LINE, PLOT THESE TWO VALUES
AS OPEN POINTS BECAUSE OF THE INEQUALITY
SYMBOL AND THEN TEST X VALUES IN EACH
OF THESE THREE INTERVALS TO SEE WHICH SATISFY
THE ORIGINAL INEQUALITY. SO LET’S TEST X EQUALS -1. LET’S SAY 1 AND LET’S SAY 4. IF WE LET X EQUAL 4, WE’D HAVE
4 SQUARED IS 16 MINUS 12. THAT’S 4,
WHICH IS GREATER THAN ZERO. SO THIS INTERVAL IS TRUE. LET’S GO AHEAD AND GRAPH IT
APPROACHING POSITIVE INFINITY. WHEN X IS 1,
WE’D HAVE 1 MINUS 3. THAT’S -2, WHICH IS
NOT GREATER THAN ZERO. SO THIS IS FALSE. AND THEN NEXT IS -1. WE’D HAVE POSITIVE 1
MINUS 3 TIMES -1. THAT BECOMES 1 PLUS 3 WHICH IS
4, WHICH IS GREATER THAN ZERO. SO THIS IS TRUE. SO THIS INTERVAL IS ALSO
A PART OF THE DOMAIN APPROACHING NEGATIVE INFINITY. SO THE DOMAIN
USING INEQUALITIES WOULD BE X IS LESS THAN ZERO
OR X IS GREATER THAN 3. WE’RE USING INTERVAL NOTATION. WE’D HAVE THE INTERVAL FROM
NEGATIVE INFINITY TO ZERO, A UNION, 3 TO INFINITY. AGAIN, IT DOES NOT INCLUDE
THESE END POINTS THEREFORE WE HAVE
OPEN INTERVALS ON 3 AND ZERO. OKAY. THANK YOU FOR WATCHING.  

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