- WE WANT TO EVALUATE THE GIVEN LOGS
GIVEN LOG X = 2, LOG Y = 4, AND LOG Z = 7.
SO TO DO THIS WE'LL EXPAND THESE LOGARITHMS
AS MUCH AS POSSIBLE
AND THEN PERFORM SUBSTITUTION FROM THE GIVEN INFORMATION
TO EVALUATE THESE LOGARITHMS.
SO FOR OUR FIRST EXAMPLE, SINCE WE HAVE A QUOTIENT HERE
WE CAN WRITE THIS AS A DIFFERENCE OF TWO LOGS
USING THIS QUOTIENT PROPERTY HERE
SO WE'LL HAVE THE LOG OF THE NUMERATOR
MINUS THE LOG OF DENOMINATOR.
SO THIS IS EQUAL TO COMMON LOG OF X SQUARED - THE COMMON LOG
OF Y Z TO THE 3rd.
NOTICE HOW WE STILL HAVE A PRODUCT HERE
SO NOW WE CAN EXPAND THIS TO A SUM OF TWO LOGS
USING THE PRODUCT PROPERTY HERE.
BUT WE NEED TO BE CAREFUL
BECAUSE WE'RE SUBTRACTING THIS LOGARITHM
SO WE'LL HAVE TO SUBTRACT THE EXPANSION OF THIS LOG.
WE HAVE COMMON LOG OF X SQUARED -
AND THEN IN PARENTHESES WE'LL HAVE COMMON LOG OF Y
+ COMMON LOG OF Z TO THE 3rd.
AND NOW WE CAN DISTRIBUTE THIS -
OR WE CAN THINK OF DISTRIBUTING A -1,
NOTICE HOW WE'LL HAVE - LOG Y - LOG Z TO THE 3rd.
SO AGAIN WE'LL HAVE LOG X SQUARED - LOG Y - LOG Z
TO THE 3rd.
NOW TO EXPAND THIS ONE MORE TIME
WE'LL APPLY THIS POWER PROPERTY OF LOGARITHMS
SO WE'LL MOVE THE EXPONENT TO THE FRONT
SO IT BECOMES THE COEFFICIENT.
SO I'LL MOVE THIS TO THE FRONT AND THIS TO THE FRONT.
SO WE'LL HAVE 2 LOG X - LOG Y - 3 LOG Z.
AND NOW WE CAN PERFORM SUBSTITUTION
FOR LOG X, LOG Y, LOG Z TO EVALUATE THIS.
SO LOG X IS EQUAL TO 2.
SO WE HAVE 2 x 2 - LOG Y IS EQUAL TO 4 - 3 x LOG Z
AND LOG Z IS EQUAL TO 7.
SO WE HAVE 4 - 4 THAT'S 0 - 21.
SO THIS IS -21.
FOR OUR SECOND EXAMPLE,
WE FIRST WANT TO WRITE THE NUMBER PART OF THE LOG
IN EXPONENT FORM.
REMEMBER WE'RE GOING TO HAVE THE EXPONENT
DIVIDED BY THE INDEX FOR EACH VARIABLE.
SO WE'LL HAVE THE COMMON LOG OF X TO THE POWER OF 9
DIVIDED BY 3,
THAT'S X TO THE 3rd.
Y TO THE POWER OF 6 DIVIDED BY 3 THAT'S 2,
SO Y SQUARED
AND THEN Z TO THE POWER OF 2 DIVIDED BY 3 WHICH IS 2/3
AND NOW WE CAN EXPAND THIS.
WE HAVE A PRODUCT HERE AND A PRODUCT HERE.
SO IN THIS CASE WE'LL HAVE A SUM OF 3 LOGS
USING THE PRODUCT PROPERTY HERE.
SO WE'RE GOING TO HAVE THE LOG OF X TO THE 3rd
+ THE LOG OF Y SQUARED + THE LOG OF Z TO THE 2/3
AND AGAIN NOW WE'LL APPLY THE POWER PROPERTY OF LOGARITHMS
SO OUR EXPONENTS WILL BE MOVED
SO THEY BECOME THE COEFFICIENTS.
SO WE'LL HAVE 3 LOG X
AND THESE ARE ALL COMMON LOGS + 2 LOG Y,
THIS WOULD BE + 2/3 LOG Z.
AND NOW I'LL PERFORM SUBSTITUTION
FOR LOG X, LOG Y, AND LOG Z
BASED UPON THE GIVEN INFORMATION.
SO WE'LL HAVE 3 x LOG X WHICH IS EQUAL TO 2, + 2 x LOG Y
WHICH IS EQUAL TO 4 + 2/3 x LOG Z WHICH IS EQUAL TO 7.
SO I'M GOING TO GO AHEAD AND WRITE THIS AS 7/1
SO IT LOOKS LIKE WE'LL HAVE 6 + 8 THAT'S 14
+ THIS WILL BE 14/3.
WELL 14 IS EQUAL TO 42/3.
LOOKS LIKE WE HAVE 56/3.
SO FOR THE MOST PART
THIS IS JUST MORE PRACTICE EXPANDING LOGARITHMS
BUT THEN SINCE WE ARE GIVEN THE VALUES OF SPECIFIC LOGS
WE CAN EVALUATE THE GIVEN LOG EXPRESSIONS.