- WE WANT TO EVALUATE
THE GIVEN LOG EXPRESSIONS
USING THE CHANGE OF BASE FORMULA
FOR LOG RHYTHMS
AND IF NEEDED, WE WANT TO
ROUND TO 4 DECIMAL PLACES.
THE CHANGE OF BASE FORMULA
GIVES US A NICE CONVENIENT WAY
TO EVALUATE ANY LOG RHYTHM
ON A CALCULATOR.
IT STATES THAT LOG BASED B OF X
IS EQUAL TO COMMON LOG X
DIVIDED BY COMMON LOG B
WHERE X IS THE NUMBERED PART
OF THE LOG
AND B IS THE BASE OF THE LOG.
IT'S ALSO EQUAL TO NATURAL LOG X
DIVIDED BY NATURAL LOG B.
NOW, I HAVE GIVEN THE CHANGE
OF BASE FORMULA
ONLY USING THE COMMON LOG
AND NATURAL LOG,
BUT IT'S ACTUALLY TRUE
FOR THE LOG OF ANY BASE,
SO IN GENERAL, WE CAN SAY THAT
LOG BASE B OF X
IS EQUAL TO LOG BASE A OF X
DIVIDED BY LOG BASE A OF--
BUT AGAIN, THE REASON I ONLY
GAVE IT IN THESE TWO FORMS
IS BECAUSE THESE ARE
THE TWO LOGS
THAT WE CAN FIND
ON ANY CALCULATOR.
SO, TO EVALUATE
LOG BASED 3 OF 9,
THIS IS GOING TO BE EQUAL TO
THE COMMON LOG OF THE NUMBER 9
DIVIDED BY THE COMMON LOG
OF THE BASE
OR IF I WANTED TO,
WE COULD USE NATURAL LOG,
NATURAL LOG 9
DIVIDED BY NATURAL LOG 3.
AND NOW, WE'LL GO
TO THE CALCULATOR
AND WE'LL JUST TYPE IN COMMON
LOG 9 DIVIDED BY COMMON LOG 3.
THIS IS EQUAL TO 2, BUT TO SHOW
WE CAN ALSO USE NATURAL LOG,
LET'S DO NATURAL LOG 9
DIVIDED BY NATURAL LOG 3.
NOTICE THE RESULT
IS EXACTLY THE SAME.
THIS IS EQUAL TO 2, SO THIS LOG
EXPRESSION IS EQUAL TO 2.
WE DIDN'T ACTUALLY NEED
TO CHANGE THE BASE FORMULA
TO DO THIS.
IF WE WANT TO CHECK THIS,
NOTICE THAT 3 RAISED
TO THE POWER OF 2 IS EQUAL TO--
SO, REMEMBER WHEN WE EVALUATE
A LOG RHYTHM,
WE'RE ACTUALLY FINDING
AN EXPONENT
IN WHICH WE CAN RAISE
THE BASE TO--
TO GET THE NUMBERED PART
OF THE LOG.
FOR OUR SECOND EXAMPLE, LET'S GO
AHEAD AND JUST USE COMMON LOG.
SO, THIS IS EQUAL
TO THE COMMON LOG OF 62,
THE NUMBER PART OF THE LOG
DIVIDED BY THE COMMON LOG
OF THE BASE, WHICH IS 5.
IN THIS ONE, WE ARE GOING
TO HAVE TO ROUND,
SO GO BACK TO THE CALCULATOR.
WE'LL PRESS LOG 62
DIVIDED BY LOG 5, PRESS ENTER.
THIS IS APPROXIMATELY 2.5643.
WHICH IF YOU WANT TO CHECK THIS,
THIS MEANS 5
RAISED TO THE POWER OF 2.5643,
SHOULD BE APPROXIMATELY 62.
LET'S GO AHEAD AND CHECK THAT.
SO, WE'RE GOING TO TAKE 5 AND
RAISE IT TO OUR ROUNDED POWER,
SO IT WILL BE OFF A LITTLE BIT,
2.5643
AND WE DID ROUND DOWN,
THAT'S WHY THE VALUE
IS JUST A LITTLE BIT UNDER 62.
LET'S GO AHEAD AND TAKE A LOOK
AT ONE MORE.
LOG BASE 2 OF 15
IS GOING TO BE EQUAL TO
COMMON LOG OF THE NUMBER 15
DIVIDED BY THE COMMON LOG OF 2.
SO, WE HAVE COMMON LOG 15
DIVIDED BY COMMON LOG 2
IS EQUAL TO APPROXIMATELY 3.9069
AND LET'S GO AHEAD
AND SHOW NATURAL LOG AS WELL.
NATURAL LOG 15
DIVIDED BY NATURAL LOG 2
WILL GIVE US THE SAME RESULT.
SO AGAIN, 3.9069,
WHICH MEANS, IF WE TAKE 2
AND RAISE IT TO THIS POWER,
WE SHOULD GET APPROXIMATELY 15.
LET'S GO AHEAD AND CHECK THAT.
WE DID ROUND UP,
THEREFORE, THE VALUE
IS SLIGHTLY LARGER THAN 15,
BUT THIS DOES VERIFY
WE DID THIS CORRECTLY.
