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- NOW WE'LL LOOK AT THREE EXAMPLES
OF EVALUATING LOG FUNCTIONS
USING THE CHANGE OF BASE FORMULA.
AND FOR THESE EXAMPLES WE'LL BE USING THE COMMON LOG
WHEN USING THE CHANGE OF BASE FORMULA.
SO THIS FIRST ROW IS AN EXAMPLE ALREADY WORKED OUT FOR US.
SO OUR FIRST FUNCTION IS H OF X = LOG BASE 7 OF X.
WE WANT TO DETERMINE H OF 5/8.
SO FOR H OF 5/8 THE INPUT IS 5/8,
SO WE'LL SUBSTITUTE 5/8 FOR X GIVING US LOG BASE 7 OF 5/8.
TO ENTER THIS INTO THE COMPUTER, IF WE CLICK IN THE ANSWER CELL,
A SMALL YELLOW ARROW WILL APPEAR.
IF WE CLICK ON THAT SMALL YELLOW ARROW,
THIS MATH PALETTE APPEARS.
IF WE CLICK ON THE FUNCTIONS TAB,
WE CAN USE LOG BASE N TO ENTER OUR LOGARITHM.
WHEN CLICK ON LOG BASE N WE CAN CHANGE THIS BASE TO 7.
AND THEN WE CAN ALSO USE THIS FRACTION TOOL
TO ENTER THE FRACTION.
NOW, FOR THE NEXT STEP WE WANT TO WRITE THIS AS A QUOTIENT
USING THE CHANGE OF BASE FORMULA GIVEN HERE.
AND BECAUSE WE'RE ASKED TO USE LOG BASE INNER COMMON LOG,
WE'LL USE THE FACT THAT LOG BASE B OF X = LOG X DIVIDED BY LOG B
WHERE WE'RE ALWAYS TAKING THE LOG OF THE NUMBER
AND THE NUMERATOR
AND THE LOG OF THE BASE IN THE DENOMINATOR.
AND THE CHANGE OF BASE FORMULA IS TRUE FOR ANY BASE LOGARITHM.
SO SOMETIMES WE'LL SEE THIS WRITTEN USING NATURAL LOG X
DIVIDED BY NATURAL LOG B.
AND THESE ARE THE TWO MOST COMMON LOGS USED
WHEN USING THE CHANGE OF BASE FORMULA
BECAUSE THESE ARE THE TWO LOGARITHMS ON THE CALCULATOR.
SO USING COMMON LOGS, WE CAN SAY THAT THIS LOGARITHM
IS EQUAL TO LOG OR COMMON LOG 5/8
DIVIDED COMMON LOG 7.
AND AGAIN, TO ENTER THIS INTO THE COMPUTER,
WE WOULD CLICK IN THE ANSWER CELL,
CLICK ON THE SMALL YELLOW ARROW.
THEN FROM THE GENERAL TAB USE THE FRACTION TOOL HERE,
THEN CLICK ON THE FUNCTIONS TAB,
AND THEN USE THIS OPTION HERE FOR COMMON LOG.
AND NOW WE'LL EVALUATE THIS ON THE CALCULATOR,
SO WE HAVE LOG 5/8, MAKE SURE WE CLOSE THE PARENTHESIS HERE,
DIVIDED BY LOG 7,
CLOSE PARENTHESIS AND ENTER.
BUT I ALSO WANT TO SHOW WE CAN GET THE SAME VALUE
USING NATURAL LOGS.
SO IF I PRESS NATURAL LOG 5/8 DIVIDED BY NATURAL LOG 7,
THE RESULT IS THE SAME.
SO WE CAN ACTUALLY USE ANY BASE WE WANT
WHEN USING THE CHANGE OF BASE FORMULA.
TO THREE DECIMAL PLACES THIS WILL BE APPROXIMATELY -0.242
BECAUSE THIS 5 IN THE FOURTH DECIMAL PLACE
INDICATES TO ROUND UP.
NEXT WE HAVE P OF T = 12 x LOG BASE 6 OF T.
WE WANT TO DETERMINE P OF 24.
SO THAT WOULD BE 12 x LOG BASE 6 OF 24.
AND WHEN APPLYING THE CHANGE OF BASE FORMULA,
WE'RE ONLY APPLYING IT TO LOGARITHM, NOT THE 12.
AND SINCE THIS MEANS 12 x LOG BASE 6 OF 24,
WE'D HAVE 12 x THE COMMON LOG 24
DIVIDED BY COMMON LOG 6.
AND NOW WE'LL EVALUATE THIS ON THE CALCULATOR.
SO WE'D HAVE 12--
AGAIN, I'M GOING TO PUT THIS WHOLE FRACTION IN PARENTHESIS.
SO OPEN PARENTHESIS, LOG 24, DIVIDED BY LOG 6.
CLOSE PARENTHESIS FOR THE LOGARITHM,
AND CLOSE PARENTHESIS FOR THE FRACTION, AND THEN ENTER.
SO P OF 24 IS APPROXIMATELY 21.284.
NOTICE 4 IN THE FOURTH DECIMAL PLACE INDICATES TO ROUND DOWN.
AND FOR THE LAST EXAMPLE WE HAVE F OF X = 14 + LOG BASE 3 OF X.
AND SO FOR F OF 193 WE'D HAVE 14 + LOG BASE 3 OF 193.
AND NOW APPLYING THE CHANGE OF BASE FORMULA,
THIS WOULD BE EQUAL TO 14 + THE COMMON LOG 193
DIVIDED BY COMMON LOG OF 3.
GOING BACK TO THE CALCULATOR,
WE'D HAVE 14 + COMMON LOG 193 DIVIDED BY COMMON LOG 3.
TO THREE DECIMAL PLACES THIS IS APPROXIMATELY 18.790.
I HOPE YOU FOUND THIS HELPFUL.