- WE WANT TO EVALUATE
THE GIVEN LOG EXPRESSION
IF NATURAL LOG X EQUALS 2
AND NATURAL LOG Y EQUALS 3
AND NATURAL LOG Z EQUALS 5.
SO, WE WANT TO EXPAND THIS
EXPRESSION AS MUCH AS POSSIBLE
USING THE PROPERTIES
OF LOGARITHMS GIVEN HERE
AND THEN EVALUATE
USING THE GIVEN INFORMATION.
ONE THING WE HAVE TO BE
CAREFUL ABOUT HERE THOUGH
IS TAKING A LOOK
AT THIS SECOND LOGARITHM,
NOTICE HOW WE HAVE
NATURAL LOG X DIVIDED BY Y
THE NEGATIVE 1
RAISED TO THE SECOND POWER.
SO, WHILE IT MIGHT BE TEMPTING
TO TRY TO APPLY THE POWER
PROPERTY OF LOGARITHMS
GIVEN HERE,
WHERE IF WE HAVE LOG X
TO THE N,
THIS IS EQUAL TO N
TIMES LOG X.
BUT THIS IS ONLY TRUE WHEN THE
EXPONENT N IS ATTACHED TO X
WITH THE NUMBER PART
OF THE LOGARITHM.
THIS PROPERTY DOES NOT APPLY
IF THE ENTIRE LOGARITHM IS
BEING RAISED TO AN EXPONENT,
WHICH IS WHAT WE HAVE.
SO, WE'LL HAVE TO EXPAND
THE LOGARITHM INSIDE
AND THEN END UP
SQUARING THE RESULT.
BUT LOOKING AT THE FIRST
NATURAL LOG,
WE CAN APPLY THE POWER
PROPERTY OF LOGARITHMS
BECAUSE WE HAVE Z RAISED
TO THE POWER OF NEGATIVE 1.
SO, WE CAN REWRITE THIS
FIRST LOGARITHM AS NEGATIVE 1
TIMES NATURAL LOG Z.
SO, WE HAVE NEGATIVE 1,
NATURAL LOG Z.
AND NOW FOR THE SECOND SET
OF PARENTHESES,
WHICH ARE STILL GONNA BE
SQUARED,
WE WANT TO EXPAND NATURAL LOG
X DIVIDED BY Y
RAISED TO THE POWER
OF NEGATIVE 1.
NOW, IF WE WANTED TO,
WE COULD REWRITE THIS FRACTION
USING POSITIVE EXPONENTS,
MEANING, IF WE HAVE X
DIVIDED BY Y THE NEGATIVE 1,
IF WE WERE TO MOVE Y
THE NEGATIVE 1
UP TO THE NUMERATOR,
IT WOULD CHANGE THE SIGN
OF THE EXPONENT.
SO, THIS IS JUST EQUAL
TO X TIMES Y.
SO, WE COULD WRITE THIS
AS NATURAL LOG XY
AND THEN EXPAND IT,
BUT LET'S GO AHEAD
AND LEAVE IT LIKE IT IS
AND EXPAND THAT
USING THE QUOTIENT PROPERTY
OF LOGARITHMS
WHERE IF WE ADD THE LOG
OF A QUOTIENT
THAT'S EQUAL TO LOG X
MINUS LOG Y.
SO, IN THIS CASE,
WE'D HAVE NATURAL LOG X
MINUS NATURAL LOG Y
TO THE POWER OF NEGATIVE 1.
AND NOW FOR THE LAST STEP,
WE CAN APPLY THE POWER
PROPERTY OF LOGARITHMS HERE
WHERE WE CAN MOVE THE EXPONENT
OF NEGATIVE 1 TO THE FRONT.
NOTICE THAT IF WE DID THAT,
WE'D HAVE MINUS NEGATIVE 1
NATURAL LOG Y,
BUT SUBTRACTING A NEGATIVE IS
THE SAME AS ADDING A POSITIVE.
SO, LET'S GO AHEAD AND MAKE
THIS AS NEGATIVE NATURAL LOG Z
AND THEN WE HAVE
NATURAL LOG X,
AGAIN PLUS NATURAL LOG Y
AND ALL THIS IS STILL SQUARED.
AND NOW IN THIS FORM,
WE'LL PERFORM SUBSTITUTION
FOR NATURAL LOG X,
NATURAL LOG Y,
AND NATURAL LOG Z.
SO, WE WOULD HAVE
NEGATIVE NATURAL LOG Z.
NATURAL LOG Z IS 5.
SO, THIS IS NEGATIVE 5
AND THEN HERE, WE HAVE
NATURAL LOG X, WHICH IS 2,
PLUS NATURAL LOG Y,
WHICH IS 3,
BUT THIS IS SQUARED.
SO, WE HAVE NEGATIVE 5 TIMES.
THIS WOULD BE 5 SQUARED, WHICH
IS JUST NEGATIVE 5 TIMES 25.
SO, OUR EXPRESSION
IS EQUAL TO NEGATIVE 125.
I HOPE YOU FOUND THIS HELPFUL.
