Tip:
Highlight text to annotate it
X
- WE'RE GOING TO COMBINE THE LOG EXPRESSIONS
INTO A SINGLE LOG BY USING THE LOG PROPERTIES
GIVEN HERE BELOW IN RED.
AND SINCE WE'RE COMBINING LOGARITHMS,
WE'LL BE USING THESE PROPERTIES IN THIS DIRECTION.
LOOKING AT THESE FIRST 2 PROPERTIES,
NOTICE THAT IN ORDER TO COMBINE A SUM OR DIFFERENCE
OF 2 LOGARITHMS INTO A SINGLE LOG,
THE COEFFICIENTS OF THE LOGARITHMS MUST BE 1.
SO IN THESE EXAMPLES, THE FIRST STEP WILL BE TO APPLY
THE POWER PROPERTY OF LOGARITHMS
WHICH STATES WE CAN MOVE THE COEFFICIENT OF THE LOGARITHM
INTO THE POSITION OF THE EXPONENT
ON THE NUMBER PART OF THE LOG.
SO FOR 5 NATURAL LOG X
+ 1/2 NATURAL LOG Y - 7 NATURAL LOG Z,
OUR FIRST STEP WILL BE TO MOVE THIS 5 TO THE EXPONENT
ON THE X,
THE 1/2 TO THE EXPONENT ON THE Y,
AND THE 7 TO THE EXPONENT ON THE Z.
SO WE'LL HAVE NATURAL LOG X TO THE 5th
+ NATURAL LOG Y TO THE 1/2 - NATURAL LOG Z TO THE 7th.
NOW THE NEXT STEP, WE'LL COMBINE THESE 2 LOGS.
AGAIN THEY'RE BOTH NATURAL LOGS.
WE HAVE A SUM,
SO WE CAN APPLY THIS PART PROPERTY OF LOGARITHMS.
BECAUSE WE HAVE A SUM, WE CAN COMBINE THEM INTO A SINGLE LOG
BY MULTIPLYING THE NUMBER PART OF THE LOGARITHMS.
SO THIS IS EQUAL TO THE NATURAL LOG OF X
TO THE 5th x Y TO THE 1/2,
AND WE STILL HAVE - NATURAL LOG Z TO THE 7th.
AND NOW, WE CAN COMBINE THESE 2 LOGARITHMS.
AGAIN, THEY'RE BOTH NATURAL LOGS.
THE COEFFICIENTS ARE 1 AND WE HAVE A DIFFERENCE,
SO WE CAN COMBINE THEM INTO A SINGLE LOG
BY FORMING THIS FRACTION HERE.
NOTICE HOW THE NUMBER PART OF THE LOG THAT WE'RE SUBTRACTING
ENDS UP IN THE DENOMINATOR OF THE FRACTION.
SO THIS IS EQUAL TO THE NATURAL LOG OF X TO THE 5th,
Y TO THE 1/2 DIVIDED BY Z TO THE 7th.
IF WE WANTED TO, WE COULD REWRITE Y TO THE 1/2
AS A SQUARE ROOT OF Y.
WE'D HAVE THE NATURAL LOG OF X TO THE 5th,
SQUARE ROOT Y, ALL OVER C TO THE 7th.
AND NOW, WE HAVE THE GIVEN LOG EXPRESSION
AS A SINGLE LOGARITHM.
LET'S TAKE A LOOK AT OUR SECOND EXAMPLE.
AGAIN, THE FIRST STEP WILL BE TO APPLY
THE POWER PROPERTY OF LOGARITHMS.
SO WE'LL MOVE THIS 4 TO THE EXPONENT ON THE 2,
WE'LL MOVE THIS 2 TO THE EXPONENT ON THE 3,
AND WE'LL LEAVE LOG 4 AS IS.
SO WE CAN WRITE THIS AS THE COMMON LOG OF 2 TO THE 4th
- THE COMMON LOG OF 3 TO THE 2nd - LOG 4.
NOW HERE, WE COULD EVALUATE 2 TO THE 4th, THAT WOULD BE 16,
AND 3 TO THE 2nd WOULD BE 9.
SO WE NEED TO GO AHEAD AND DO THAT.
THIS IS EQUAL TO THE LOG OF 16 - THE LOG OF 9 - THE LOG OF 4.
NOW, LET'S GO AHEAD AND TAKE THIS 2 AT A TIME.
HERE WE HAVE A DIFFERENCE OF 2 LOGARITHMS WITH THE SAME BASE
AND A COEFFICIENT OF 1,
SO WE CAN COMBINE THESE INTO A SINGLE LOG
BY FORMING A FRACTION
WITH THE NUMBER PARTS OF THE LOGARITHMS.
SO THIS IS EQUAL TO THE COMMON LOG OF 16
DIVIDED BY 9 - COMMON LOG OF 4.
AND NOW, WE CAN ALSO COMBINE THESE 2 LOGARITHMS.
AND AGAIN SINCE WE'RE SUBTRACTING LOG 4,
THE 4 WILL END UP IN THE DENOMINATOR,
OR IN THIS CASE WE MULTIPLIED BY THE 9
THAT'S ALREADY IN THE DENOMINATOR.
SO THIS IS EQUAL TO THE LOG OF 16 DIVIDED BY 9 x 4
WHICH WOULD BE THE LOG OF 16/36 WHICH DOES SIMPLIFY.
16 AND 36 DO SHARE A COMMON FACTOR OF 4.
THERE ARE 4 4s IN 16 AND THERE ARE 9 4s IN 36,
SO THIS SIMPLIFIES TO THE COMMON LOG OF 4/9.
OKAY, I HOPE THIS HELPS EXPLAIN
HOW TO COMBINE LOG EXPRESSIONS INTO SINGLE LOGS.