- A TABLET COMPUTER 
IS WORTH $1,500 NEW.
IT LOSES 12% OF ITS VALUE 
EACH YEAR.
THE VALUE CAN BE MODELED BY THE 
FUNCTION V OF T EQUALS A TIMES B
RAISED TO THE POWER OF T,
WHERE V OF T IS THE DOLLAR VALUE
AND T IS THE NUMBER OF YEARS 
SINCE THE PURCHASE.
WE WANT TO WRITE A POSSIBLE 
EXPONENTIAL EQUATION
FOR THE VALUE.
SO OUR FUNCTION WILL BE 
IN THE FORM OF V OF T.
A REPRESENTS THE INITIAL VALUE 
OF THE COMPUTER
WHICH WILL BE 1,500
AND WE'RE GOING TO HAVE SUM 
BASE B RAISED TO THE POWER OF T.
SO LET'S TALK ABOUT 
HOW WE'RE GOING TO FIND THE BASE
OF THIS EXPONENTIAL FUNCTION.
WE KNOW EACH YEAR THE COMPUTER 
LOSES 12% OF THE VALUE,
WHICH MEANS IT RETAINS 
88% OF THE VALUE.
SO AFTER THE FIRST YEAR, THE 
VALUE WOULD BE 1,500 TIMES 0.88.
THIS 0.88 REPRESENTS THE 88% 
OF THE VALUE THAT IT RETAINS
AFTER THE FIRST YEAR.
AFTER THE SECOND YEAR
THE COMPUTER LOSES ANOTHER 12% 
OF THE VALUE OFF THIS AMOUNT
OR RETAINS 88% 
OF THIS VALUE HERE.
SO AFTER THE SECOND YEAR IT 
WOULD BE 1,500 TIMES 0.88
TIMES ANOTHER 0.88.
AGAIN, THIS SECOND 0.88 
REPRESENTS THE AMOUNT OF VALUE
RETAINED AFTER THE SECOND YEAR
AND THIS PATTERN 
IS GOING TO CONTINUE.
AFTER THE THIRD YEAR WE'LL HAVE 
1,500 TIMES 0.88 TIMES 0.88
TIMES ANOTHER 0.88.
SO THE BASE OF OUR EXPONENTIAL 
FUNCTION IS GOING TO BE 0.88
WHICH IS THE PERCENT OF VALUE 
THE COMPUTER RETAINS
AFTER EACH YEAR.
SO TO FIND THE VALUE 
AFTER THREE YEARS
WE NEED TO DETERMINE VIA 3.
SO WE'LL SUBSTITUTE 3 FOR T
AND EVALUATE THIS 
ON THE CALCULATOR.
SO WE'LL HAVE 1,500 TIMES 0.88 
RAISED TO THE POWER OF 3.
SO THE VALUE OF THE COMPUTER 
IS APPROXIMATELY 1,022.21
AFTER THREE YEARS.
AND NOW FOR THE LAST QUESTION,
WHEN WILL THE VALUE BE HALF 
OF ITS ORIGINAL VALUE?
WELL, 
THE STARTING VALUE IS 1,500
SO HALF OF THIS WOULD BE 750.
SO WE WANT TO SET D OF T TO 750 
AND THEN SOLVE FOR T.
SO WE'LL HAVE 750 EQUALS 1,500 
TIMES 0.88
RAISED TO THE POWER OF T.
SO WE WANT TO ISOLATE
THE EXPONENTIAL PART 
OF THIS EQUATION
SO WE'LL DIVIDE BOTH SIDES 
BY 1,500.
THIS WILL BE ONE 750 
DIVIDED BY 1,500 EQUALS 1/2
OR 0.5 EQUALS 0.88 
RAISED TO THE POWER OF T.
AND NOW WE CAN USE LOGARITHMS 
TO SOLVE THIS FOR T.
IF WE TAKE THE NATURAL LOG 
OF BOTH SIDES OF THE EQUATION,
ON THE RIGHT SIDE WE CAN USE 
THE POWER PROPERTY OF LOGARITHMS
TO MOVE THIS T TO THE FRONT,
GIVING US NATURAL LOG OF 0.5 
EQUALS T TIMES NATURAL LOG 0.88.
NOW, TO SOLVE THIS EQUATION 
FOR T
WE CAN DIVIDE BOTH SIDES 
BY NATURAL LOG 0.88.
NOTICE HERE THIS SIMPLIFIES TO A 
1 SO THIS QUOTIENT WILL EQUAL T.
SO WE'LL GO BACK 
TO THE CALCULATOR.
NATURAL LOG 0.5 
DIVIDED BY NATURAL LOG 0.88
WILL GIVE US THE VALUE OF T.
WHEN ROUNDED TO THE NEAREST 10TH
THIS WILL BE APPROXIMATELY 
5.4 YEARS.
SO AGAIN THE VALUE WILL BE HALF 
OF THE ORIGINAL VALUE
AFTER 5.4 YEARS
OR APPROXIMATELY 5.4 YEARS.
