- IN THIS QUESTION,
WE'RE ASKED TO EVALUATE THE NATURAL LOGARITHM GIVEN HERE
IF NATURAL LOG X EQUALS 2, NATURAL LOG Y EQUALS 3,
AND NATURAL LOG Z EQUALS 5.
TO DO THIS, WE WILL EXPAND THIS LOGARITHM
AS MUCH AS POSSIBLE
AND THEN PERFORM SUBSTITUTION FOR NATURAL LOG X,
NATURAL LOG Y AND NATURAL LOG Z.
SO IF WE TAKE A LOOK AT THE GIVEN LOGARITHM,
NOTICE HOW THIS QUOTIENT HERE INVOLVES NEGATIVE EXPONENTS,
SO IF WE WANTED TO, WE COULD REWRITE THIS FRACTION HERE
USING POSITIVE EXPONENTS.
FOR EXAMPLE, IF WE HAVE X TO THE POWER OF -2,
DIVIDED BY Y TO THE POWER OF -3, Z TO THE 4TH,
IF WE MOVE X TO THE -2 TO THE DENOMINATOR
IT WOULD CHANGE THE SIGN OF THE EXPONENT TO POSITIVE 2
AND IF WE MOVED Y TO THE -3 UP TO THE NUMERATOR,
IT WOULD CHANGE THE EXPONENT TO POSITIVE 3.
SO THIS FRACTION IS EQUAL TO Y TO THE 3RD IN THE NUMERATOR
AND THEN THE DENOMINATOR WOULD BE X TO THE 2ND,
Z TO THE 4TH.
SO WE COULD RELATE THE LOGARITHM
USING THIS FRACTION INSTEAD
BUT LET'S GO AHEAD AND USE THE LOGARITHM IN THE CURRENT FORM.
SO BECAUSE WE HAVE THE NATURAL LOG OF A QUOTIENT,
WE CAN WRITE THIS AS A DIFFERENCE OF TWO LOGARITHMS
USING THE QUOTIENT PROPERTY OF LOGARITHMS HERE.
OR IF WE HAD THE LOG OF A QUOTIENT,
WE CAN WRITE THE LOG OF THE NUMERATOR
MINUS THE LOG OF THE DENOMINATOR.
SO THIS IS EQUAL TO NATURAL LOG, X TO THE POWER OF -2,
MINUS NATURAL LOG OF Y TO THE -3 TIMES Z TO THE 4TH.
BECAUSE THE SECOND LOG IS THE NATURAL LOG OF A PRODUCT
WE CAN EXPAND THIS AS THE SUM OF TWO LOGS
USING THE PRODUCT PROPERTY OF LOGARITHMS GIVEN HERE.
BUT WE HAVE TO BE CAREFUL HERE
BECAUSE IF WE'RE SUBTRACTING THIS LOG,
WE ALSO HAVE TO SUBTRACT THE EXPANSION
OR THE SUM OF THE TWO LOGS.
SO THIS WOULD BE EQUAL TO NATURAL LOG X TO THE -2, MINUS
AND THEN IN BRACKETS OR PARENTHESES
WE'D HAVE NATURAL LOG Y TO THE -3
PLUS NATURAL LOG Z TO THE 4TH.
NOW WE CAN APPLY THE POWER PROPERTY OF LOGARITHMS
GIVEN HERE WHERE WE CAN TAKE THE EXPONENT N
AND MOVE IT TO THE FRONT
SO WE HAVE A PRODUCT OF N TIMES THE LOG.
SO WE CAN TAKE THE EXPONENT OF -2,
MOVE IT TO THE FRONT SO IT'S THE COEFFICIENT,
TAKE THIS -3 AND MOVE IT TO THE FRONT
AND MOVE THIS 4 TO THE FRONT.
SO THIS WOULD GIVE US -2 NATURAL LOG X
MINUS THE QUANTITY -3 NATURAL LOG Y
AND THEN PLUS 3 NATURAL LOG Z.
BUT BEFORE WE PERFORM THE SUBSTITUTION
LET'S GO AHEAD AND CLEAR THESE BRACKETS
SO BECAUSE WE HAVE SUBTRACTION HERE
WE CAN THINK OF DISTRIBUTING A -1
WHICH WOULD GIVE US -2 NATURAL LOG X,
AND THEN -1 TIMES -3 IS POSITIVE 3.
SO PLUS 3 NATURAL LOG Y
AND THEN -1 TIMES POSITIVE 4 WOULD BE -4.
SO WE HAVE -4 NATURAL LOG Z.
AND NOW ON THIS FORM WE'LL SUBSTITUTE THE VALUES
FOR NATURAL LOG X, NATURAL LOG Y AND NATURAL LOG Z.
SO WE WOULD HAVE -2 TIMES NATURAL LOG X
WHICH IS EQUAL TO 2 PLUS 3 TIMES NATURAL LOG Y.
NATURAL LOG Y IS 3, MINUS 4 TIMES NATURAL LOG Z
WHERE NATURAL LOG Z IS POSITIVE 5.
SO THIS WOULD BE -4 PLUS 9 MINUS 20.
WELL, -4 PLUS 9 IS 5, 5 MINUS 20 IS EQUAL TO -15.
SO THE GIVEN NATURAL LOG IS EQUAL TO -15.
I HOPE YOU FOUND THIS HELPFUL.