- 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.