- NOW, WE'LL TAKE A LOOK
AT SOLVING
SEVERAL EXPONENTIAL EQUATIONS
USING THE PROPERTIES
OF EXPONENTS.
SO LOOKING AT OUR FIRST EXAMPLE,
WE HAVE 2
RAISED TO THE POWER OF X = 64.
IF WE COULD WRITE THE RIGHT SIDE
OF THE EQUATION
AS 2 RAISED TO ITS OWN POWER,
WE COULD THEN SOLVE FOR X.
IN GENERAL IF WE HAVE "A"
RAISED TO THE POWER OF M
= "A" RAISED TO THE POWER OF N,
THEN M IS EQUAL TO N, MEANING
IF THE BASES ARE THE SAME
AND THEY'RE EQUAL TO EACH OTHER,
THE EXPONENTS
MUST ALSO BE EQUAL.
SO IF WE CAN WRITE 64
AS 2 RAISED TO ITS OWN POWER,
WE CAN SOLVE THIS EQUATION
WITHOUT USING LOGARITHMS.
AND IF WE HAVE A HARD TIME
DETERMINING IF WE CAN WRITE 64
AS 2 RAISED TO ITS OWN POWER,
IT'S OFTEN HELPFUL TO LOOK
AT THE PRIME FACTORIZATION.
LET'S GO AHEAD AND DO THAT.
64 IS EQUAL TO 8 x 8,
AND 8 IS EQUAL TO 4 x 2,
WELL, 2 IS PRIME, AND OF COURSE
4 IS EQUAL TO 2 x 2.
NOTICE HOW 64
CONTAINS 6 FACTORS OF 2
WHICH MEANS WE CAN REWRITE THIS
EQUATION AS 2 TO THE POWER OF X
= 2 TO THE POWER OF 6.
AND SINCE OUR BASES ARE THE SAME
AND THESE ARE EQUAL
TO EACH OTHER,
X MUST EQUAL 6
WHICH IS OUR SOLUTION.
LOOKING AT OUR SECOND EXAMPLE,
EVEN THOUGH THIS IS
IN FRACTION FORM,
LET'S SEE IF WE CAN WRITE 81
AS 3 RAISED TO ITS OWN POWER.
AGAIN JUST IN CASE WE NEED TO,
WE CAN WRITE OUT
THE PRIME FACTORIZATION
81 IS EQUAL TO 9 x 9,
AND OF COURSE 9
IS EQUAL TO 3 x 3,
SO 81 IS EQUAL TO
3 TO THE FOURTH.
SO WE COULD WRITE THIS AS
3 TO THE POWER OF X
= 1/3 TO THE POWER OF 4.
AND NOW, WE CAN USE
OUR PROPERTIES OF EXPONENTS
AND MOVE THIS UP
INTO THE NUMERATOR.
REMEMBER IF WE DO THIS,
IT'S GOING TO CHANGE
THE SIGN OF THE EXPONENT.
SO WE CAN NOW WRITE THIS AS
3 TO THE POWER OF X
= 3 TO THE POWER OF -4.
AND AGAIN NOW THAT THE BASES
ARE THE SAME,
THE EXPONENTS MUST BE EQUAL,
SO X IS EQUAL TO -4
WHICH IS OUR SOLUTION.
WE WILL BE LOOKING
AT SEVERAL MORE EXAMPLES,
AND YOU CAN WATCH THEM
IN SEQUENCE
BY CLICKING ON THE LINK
IN THE UPPER LEFT-HAND CORNER.
I HOPE YOU FOUND THIS HELPFUL.
