TO DEMONSTRATE THE PROPERTY 
OF LOGARITHMS,
WE'RE ASKED TO EXPAND THIS 
LOGARITHM AS MUCH AS POSSIBLE.
WE HAVE NATURAL LOG OF X 
TO THE FOURTH,
Y TO THE SECOND x THE FIFTH 
ROOT OF X TO THE THIRD,
Y TO THE FOURTH.
BUT NOTICE HOW WE HAVE X's 
OUTSIDE THE RADICAL
AND UNDERNEATH THE RADICAL,
AS WELL AS Y's 
OUTSIDE THE RADICAL
AND UNDERNEATH THE RADICAL.
SO THERE'S COUPLE WAYS 
OF DOING THIS,
BUT WHAT I'M GOING TO DO 
IS REWRITE THIS RADICAL
USING RATIONAL EXPONENTS.
SO WE CAN WRITE THIS 
AS NATURAL LOG,
LEAVE X TO THE FOURTH
AND Y TO THE SECOND ALONE 
FOR RIGHT NOW.
BUT THEN THE FIFTH ROOT OF X 
TO THE THIRD
IS THE SAME AS X 
TO THE 3/5 POWER.
AND THE FIFTH ROOT OF Y 
TO THE FOURTH
IS THE SAME AS Y 
TO THE 4/5 POWER.
NOW THAT IT'S IN THIS FORM WE 
CAN MULTIPLY X TO THE FOURTH
AND X TO THE 3/5,
AS WELL AS Y TO THE SECOND 
AND Y TO THE 4/5.
REMEMBER WHEN MULTIPLYING 
WHEN THE BASES ARE THE SAME,
WE ADD THE EXPONENTS.
SO TO MULTIPLY X TO THE FOURTH 
AND X TO THE 3/5,
WE NEED TO ADD THE EXPONENTS.
WELL, 4 + 3/5 
IS JUST 4 AND 3/5.
LET'S GO AHEAD AND CONVERT 
THIS TO AN IMPROPER FRACTION.
5 x 4 IS 20 + 3 
THAT WOULD BE 23/5.
SO WE'D HAVE NATURAL LOG OF X 
TO THE 23/5.
THEN WE'LL HAVE Y 
TO THE POWER OF 2 + 4/5.
WELL, 2 + 4/5 IS JUST 2 
AND 4/5.
CONVERT THIS 
TO AN IMPROPER FRACTION,
5 x 2 + 4 THAT'S 14/5.
SO WE HAVE Y TO THE 14/5.
NOW, NOTICE HOW WE HAVE 
NATURAL LOG OF A PRODUCT,
SO WE CAN EXPAND THIS FURTHER
USING THE PRODUCT PROPERTY 
OF LOGARITHMS GIVEN HERE.
SO THIS IS GOING TO BE = TO 
NATURAL LOG OF X TO THE 23/5 +
BECAUSE WE HAVE A PRODUCT 
NATURAL LOG Y TO THE 14/5.
AND NOW WE CAN EXPAND THIS 
ONE MORE TIME
BY USING THE POWER PROPERTY 
OF LOGARITHMS GIVEN HERE
WHERE WE HAVE LOG BASE B OF X 
RAISED TO THE POWER OF N
IS = TO N x LOG BASE B OF X.
SO WE CAN TAKE THIS EXPONENT
AND MOVE IT TO THE FRONT OF 
THE LOGARITHM HERE AND HERE.
SO IN EXPANDED FORM 
WE WOULD HAVE
23/5 NATURAL LOG X + 14/5 
NATURAL LOG Y.
THIS WOULD BE THE GIVEN 
LOGARITHM
EXPANDED AS MUCH AS POSSIBLE.
