TO DEMONSTRATE THE PROPERTIES
OF LOGARITHMS,
WE'RE ASKED TO WRITE THE SUM
OF THESE TWO LOGS
AS A SINGLE LOG
AND THE DIFFERENCE OF THESE
TWO LOGS AS A SINGLE LOG.
SO BECAUSE WE'RE TRYING
TO COMBINE THE SUM
OF TWO LOGS HERE,
WE'LL BE USING THE PRODUCT
PROPERTY OF LOGARITHMS
GIVEN HERE,
WHERE LOG BASE B OF X x Y
IS EQUAL TO LOG BASE B OF X
+ LOG BASE B OF Y.
SO IF WE WANT TO COMBINE
THE SUM OF TWO LOGS
INTO A SINGLE LOG
THERE ARE TWO THINGS
WE SHOULD NOTICE.
FIRST, THE BASES
MUST BE THE SAME
AND THEN BECAUSE WE HAVE
A SUM,
TO COMBINE THE TWO LOGS
WE'RE GOING TO MULTIPLY
THE NUMBER OF PARTS OF THE
LOGARITHMS AS A SINGLE LOG.
SO FOR LOG BASE 3
OF 4 X SQUARED Y
+ LOG BASE 3 OF 5 X
CUBED Y SQUARED
THIS WILL BE EQUAL TO LOG
BASE 3 OF 4X SQUARED Y x 5 X
TO THE 3rd Y SQUARED.
SO WE HAVE COMBINED THIS
INTO A SINGLE LOG
BUT WE DO WANT TO FIND
THIS PRODUCT HERE
SO THIS WILL EQUAL LOG BASE 3
OF 4 x 5 IS 20.
X TO THE 2nd x X TO THE 3rd
IS X TO THE 5th
AND Y TO THE 1st x Y
TO THE 2nd IS Y TO THE 3rd.
WE HAVE NOW COMBINED THIS SUM
OF TWO LOGS INTO A SINGLE LOG.
LOOKING AT THE SECOND EXAMPLE,
NOTICE HOW WE HAVE
A DIFFERENCE
SO WE'LL BE USING THE QUOTIENT
PROPERTY OF LOGARITHMS HERE.
WE HAVE LOG BASE B OF X
DIVIDED BY Y
EQUALS LOG BASE B OF X
- LOG BASE B OF Y.
SO AGAIN
WHAT WE SHOULD NOTICE HERE
IS THAT IF WE HAVE
A DIFFERENCE OF TWO LOGS
AND THE BASES ARE THE SAME,
THEN BECAUSE WE HAVE
A DIFFERENCE OF TWO LOGS
THIS TIME WE CAN COMBINE THEM
INTO A SINGLE LOG
IF WE FIND THE QUOTIENT
OF X AND Y.
SO FOR OUR SECOND EXAMPLE,
BECAUSE WE HAVE A DIFFERENCE
AND THE BASES ARE THE SAME
WE CAN WRITE THIS AS LOG
BASE 7 OF 3 X TO THE 4th
Y TO THE 3rd
DIVIDED BY 5 X TO THE 6th
Y TO THE 3rd.
SO WE HAVE COMBINED THIS
INTO A SINGLE LOG
BUT WE DO WANT TO SIMPLY THIS
FRACTION AS MUCH AS POSSIBLE.
NOTICE WE HAVE Y TO THE 3rd/Y
TO THE 3rd,
THAT SIMPLIFIES TO 1,
AND THEN WE HAVE X
TO THE 4th/X TO THE 6th,
NOTICE HOW ARE THERE
ARE TWO MORE FACTORS OF X
IN THE DENOMINATOR
SO WHEN WE SIMPLIFY THIS THE X
TO THE 4th WOULD SIMPLIFY OUT,
THE X TO THE 6th WOULD
SIMPLIFY TO X TO THE 2nd,
AND THE 3/5 DOES NOT SIMPLIFY
SO THIS WOULD BE LOG BASE 7
OF 3/5 X TO THE 2nd.
TAKING A CLOSER LOOK AT X
TO THE 4th
DIVIDED BY X TO THE 6th,
THERE ARE SEVERAL WAYS
TO SIMPLIFY THIS.
BECAUSE WE HAVE A QUOTIENT WE
COULD SUBTRACT THE EXPONENTS,
4 - 6 = -2
AND X TO THE -2 = 1/X SQUARED.
NOTICE HOW WE STILL HAVE
TWO FACTORS OF X
IN THE DENOMINATOR
AS WE DO HERE.
AS A LAST RESORT,
WE COULD JUST EXPAND THIS.
THERE'D BE 4 FACTORS OF X
ON TOP
AND 6 FACTORS OF X
ON THE BOTTOM.
EVERY X/X SIMPLIFIES TO 1, SO
THAT SIMPLIFIES TO 1, 1, 1, 1.
EITHER WAY WE END UP
WITH TWO FACTORS OF X
IN THE DENOMINATOR
OR X SQUARED
IN THE DENOMINATOR.
OKAY. I HOPE YOU FOUND
THIS HELPFUL.
