- WE WANT TO WRITE
THE GIVEN LOG EXPRESSIONS
AS A SINGLE LOG
OR COMBINE THE LOGARITHMS.
TO DO THIS WE'LL BE USING
THE PROPERTY OF LOGARITHMS
GIVEN HERE BELOW IN RED,
THE SAME ONES THAT WE USED
TO EXPAND LOGS,
BUT NOW WE'LL BE USING
THESE PROPERTIES
IN THE REVERSE ORDER.
OUR FIRST EXPRESSION
WE HAVE 2 LOG BASE 5 OF X
+ 3 LOG BASE 5 OF 2.
WE SHOULD RECOGNIZE
THAT SINCE WE HAVE THE SUM,
THIS FIRST PROPERTY
IS GOING TO APPLY,
BUT NOTICE HOW THEY'LL COMBINE
THESE LOGS INTO A SINGLE LOG.
THE COEFFICIENTS
OF THESE LOGARITHMS MUST BE
SO BEFORE WE APPLY
THIS PROPERTY,
WE'LL HAVE TO APPLY THE POWER
PROPERTY OF LOGARITHMS HERE
THAT SAYS WE COULD MOVE
THE COEFFICIENTS
INTO THE EXPONENT POSITION
OF THE NUMBER PART OF THE LOG,
WHICH MEANS WE CAN TAKE
THIS 2 HERE
AND MAKE IT THE EXPONENT
ON THE X,
AND WE CAN MOVE THIS 3 HERE
AND MAKE IT THE EXPONENT
ON THIS 2.
SO WE CAN WRITE THIS
AS LOG BASE 5 OF X TO THE 2nd
+ LOG BASE 5 OF 2 TO THE 3rd.
NOW, WE CANNOT EVALUATE
X SQUARED,
BUT WE CAN EVALUATE 2
TO THE 3rd.
2 TO THE 3rd = 8.
SO WE CAN WRITE THIS
AS LOG BASE 5 OF X SQUARED
+ LOG BASE 5 OF 8,
AND NOW, BECAUSE WE HAVE A SUM
OF TWO LOGS
WITH THE SAME BASE,
WE COULD APPLY
OUR PRODUCT PROPERTY,
WHICH MEANS WE CAN WRITE THIS
AS A SINGLE LOG
BY MULTIPLYING THE NUMBER
PARTS OF THE LOGARITHMS.
SO THIS IS EQUAL TO LOG BASE 5
OF X TO THE 2nd x 8,
WHICH WOULD JUST BE 8X
TO THE 2nd.
THIS WOULD BE THE SINGLE LOG
FOR THE GIVEN LOG EXPRESSION.
LOOKING AT OUR SECOND EXAMPLE,
AGAIN, WE DO HAVE A DIFFERENCE
NOW.
SO WE WILL APPLY THIS QUOTIENT
PROPERTY OF LOGARITHMS,
BUT AGAIN, BEFORE WE DO THIS,
RECOGNIZE THE COEFFICIENTS
OF THE LOGS MUST BE 1.
SO AGAIN, WE'LL APPLY THE
POWER PROPERTY OF LOGS FIRST.
SO WE'LL MOVE THIS COEFFICIENT
OF 2 TO THE EXPONENT ON THE 8,
AND WE'LL MOVE THIS 3
TO THE EXPONENT ON THE 2.
SO THIS IS EQUAL TO COMMON LOG
OF 8 SQUARED - THE COMMON LOG
OF 2 TO THE 3rd.
WE'LL GO AHEAD AND EVALUATE 8
TO THE 2nd AND 2 TO THE 3rd.
SO THIS IS THE LOG OF 64
- LOG OF 2 TO THE 3rd IS 8,
AND NOW BECAUSE WE HAVE
A DIFFERENCE
AND TWO LOGS WITH THE SAME
BASE WITH A COEFFICIENT OF 1,
WE CAN COMBINE THE TWO LOGS
BY CREATING THE QUOTIENT OF
THE NUMBER PARTS OF THE LOGS.
SO THIS IS EQUAL TO THE COMMON
LOG OF 64 DIVIDED BY 8,
AND THIS SIMPLIFIES AS WELL.
64 DIVIDED BY 8 = 8.
SO THIS SIMPLIFIES
TO THE COMMON LOG OF 8.
OKAY, WE'LL TAKE A LOOK
AT TWO MORE EXAMPLES
IN THE NEXT VIDEO.
