- WE WANT TO EVALUATE THE EXPRESSION
INVOLVING NATURAL LOG
GIVEN NATURAL LOG X = 2, NATURAL LOG Y = 3,
AND NATURAL LOG Z = 5.
SO WE WANT TO EXPAND THIS LOGARITHM AS MUCH AS POSSIBLE
USING OUR PROPERTIES OF LOGARITHMS,
SO THAT WE CAN THEN USE THE GIVEN VALUES
TO DETERMINE THE VALUE OF THE GIVEN EXPRESSION.
FIRST THING WE WANT TO DO IS REWRITE THE CUBE ROOT
USING A RATIONAL EXPONENT.
SO IF WE HAVE THE CUBE ROOT OF A, THE INDEX IS 3,
AND THE EXPONENT ON A WOULD BE 1.
SO TAKING THE CUBE ROOT OF A IS THE SAME
AS A TO THE 1/3 POWER.
WHICH MEANS WE CAN REWRITE THE GIVEN NATURAL LOG
AS NATURAL LOG X TO THE POWER OF 2,
Y TO THE POWER OF -4,
C TO THE POWER OF -1 RAISED TO THE 1/3 POWER.
IN THIS FORM WE HAVE POWERS RAISED TO POWERS,
SO WE CAN NOW MULTIPLY THE EXPONENTS.
JUST FOR REVIEW,
IF WE HAVE A TO THE M RAISED TO THE POWER OF N,
THIS IS = TO A TO THE POWER OF M x N.
WHICH MEANS THIS IS = TO NATURAL LOG OF X
TO THE POWER OF 2 x 1/3 THAT'S 2/3.
Y TO THE POWER OF -4 x 1/3 IS -4/3.
AND Z TO THE POWER OF -1 x 1/3 IS -1/3.
IN THIS FORM WE HAVE NATURAL LOG X TO THE 2/3
x Y TO THE -4/3 x Z TO THE -1/3.
SO BECAUSE WE HAVE THE NATURAL LOG OF A PRODUCT,
WE COULD EXPAND THIS LOGARITHM
USING OUR PRODUCT PROPERTY HERE
WHERE LOG XY = LOG X + LOG Y.
AND SINCE WE HAVE A PRODUCT INVOLVING X, Y, AND Z,
WE'LL ACTUALLY HAVE A SUM OF THREE LOGS.
THIS IS = TO NATURAL LOG OF X TO THE 2/3
+ NATURAL LOG Y TO THE -4/3,
AND + NATURAL LOG Z TO THE - 1/3.
AND NOW WE CAN APPLY THE POWER PROPERTY OF LOGARITHMS
GIVEN HERE,
OR WE CAN TAKE THE EXPONENT N AND MOVE IT TO THE FRONT,
SO WE HAVE N X THE LOGARITHM.
OR IN OUR CASE,
WE CAN MOVE THE EXPONENT OF 2/3 TO THE FRONT,
THE EXPONENT OF -4/3 TO THE FRONT,
AND THE EXPONENT OF -1/3 TO THE FRONT.
SO THIS IS = TO 2/3 NATURAL LOG X.
NOW, FOR THE NEXT LOGARITHM,
SINCE THE COEFFICIENT WOULD BE -4/3,
WE CAN JUST GO AHEAD AND WRITE -4/3 NATURAL LOG Y.
AND THE SAME THING HERE,
INSTEAD OF WRITING + -1/3 NATURAL LOG,
WE CAN JUST WRITE - 1/3 NATURAL LOG Z.
IN THIS FORM WE CAN SUBSTITUTE 2 FOR NATURAL LOG X,
3 FOR NATURAL LOG Y, AND 5 FOR NATURAL LOG Z.
SO WE WOULD HAVE 2/3
x NATURAL LOG X, WHICH IS 2 OR 2/1 - 4/3
x NATURAL LOG Y, WHICH IS 3 OR 3/1 - 1/3
NATURAL LOG Z, WHICH IS 5 OR 5/1.
I'M NOT GOING TO SIMPLIFY THIS HERE
BECAUSE NOTICE ALL THE DENOMINATORS WILL BE 3,
THEREFORE WE CAN FIND THIS DIFFERENCE.
SO HERE WE HAVE 4/3 - THIS WOULD BE 12/3, AND THEN - 5/3.
WELL, THE DENOMINATOR IS 3,
SO WE CAN GO AHEAD AND PERFORM THE SUBTRACTION, 4 - 12 IS -8,
AND -8 - 5 IS -13.
SO THE GIVEN EXPRESSION INVOLVING NATURAL LOG
IS = TO -13/3.
I HOPE YOU FOUND THIS HELPFUL.