- YOU DEPOSIT $1,000 
INTO A BANK
THAT PAYS 8% CONTINUOUS  
INTEREST.
WHAT EQUATION CAN BE USED TO 
DETERMINE THE ACCOUNT BALANCE
AFTER TWO YEARS,
WHAT IS THE ACCOUNT BALANCE 
AFTER FIVE YEARS,
AND HOW LONG WILL IT TAKE 
FOR THE INVESTMENT TO DOUBLE.
WHEN DOING THESE TYPES 
OF PROBLEMS,
IT'S IMPORTANT THAT WE PAY 
CLOSE ATTENTION
TO THE TYPE OF INTEREST.
THIS IS CONTINUOUS INTEREST,
WHICH TELLS US WE'LL BE USING 
THE FORMULA A = P x E
RAISED TO POWER OF RT.
WHERE P IS THE PRINCIPLE 
OR STARTING AMOUNT
IN THE ACCOUNT,
R IS THE ANNUAL INTEREST RATE 
EXPRESSED AS A DECIMAL,
T IS THE TIME IN YEARS, AND A 
IS THE AMOUNT AFTER T YEARS.
SO FOR THIS SITUATION, 
WE'D HAVE THE EQUATION A = P,
WHICH IS $1,000 x E 
RAISED TO THE POWER OF R x T.
WELL, R IS THE INTEREST RATE,
8% AS A DECIMAL WOULD BE 0.08 
x THE TIME IN YEARS,
WHICH IS OUR VARIABLE T.
SO THIS WOULD BE THE EQUATION 
FOR THIS ACCOUNT.
NEXT WE WANT TO KNOW WHAT 
THE ACCOUNT BALANCE WOULD BE
AFTER FIVE YEARS.
SO WE'RE GOING TO SUBSTITUTE 5 
FOR T.
SO A OF 5 WOULD BE 
= TO $1,000 x E
RAISED TO THE POWER 
OF 0.08 x 5.
AND NOW WE'LL GET 
A DECIMAL APPROXIMATION
WITH THE CALCULATOR.
SECOND NATURAL LOG BRINGS UP E 
TO THE POWER OF,
JUST TYPE IN 0.08 
OR JUST 0.08 x 5.
AND SO THE ACCOUNT BALANCE 
WOULD BE $1,491
AND APPROXIMATELY 82 CENTS.
AND NOW FOR THE LAST QUESTION,
WE WANT TO KNOW HOW LONG IT 
WOULD TAKE FOR THE INVESTMENT
TO DOUBLE.
SO THERE'S A SHORTCUT 
FOR DOUBLING TIME.
I'M GOING TO GO AHEAD 
AND WRITE THIS OUT
LIKE WE NORMALLY WOULD,
MEANING IF WE START 
WITH A $1,000,
IF WE WANT THIS TO DOUBLE, 
2 x 1,000 IS 2,000.
SO WE'LL SOLVE THIS EQUATION 
HERE WHEN WE SET A = 2,000,
WHICH MEANS WE WANT TO SOLVE 
THE EQUATION 2,000 = 1,000
x E RAISED TO THE POWER 
OF 0.08 x T.
SO WE HAVE AN EXPONENTIAL 
EQUATION,
SO WE WANT TO ISOLATE 
THE EXPONENTIAL PART.
SO WE'LL DIVIDE BOTH SIDES 
OF THE EQUATION BY 1,000.
THIS SIMPLIFIES TO 1, 
THE LEFT SIMPLIFIES TO 2.
SO FOR DOUBLING TIME 
THIS WOULD ALWAYS BE 2
BECAUSE THIS AMOUNT HERE
WOULD ALWAYS BE TWICE 
THE AMOUNT OF P HERE.
SO WE HAVE 2 = E 
RAISED TO THE POWER OF 0.08 T.
AND NOW WE'LL TAKE 
THE NATURAL LOG
ON BOTH SIDES OF THE EQUATION,
AND WE'LL APPLY THE POWER 
PROPERTY OF LOGARITHMS HERE.
SO WE CAN TAKE THIS EXPONENT 
AND MOVE IT TO THE FRONT,
AND NOW WE HAVE NATURAL LOG 2 
= 0.08T x NATURAL LOG E.
BUT NATURAL LOG E, 
REMEMBER IS LOG BASE E.
AND SINCE E TO THE FIRST POWER 
IS = TO E THIS IS = TO 1,
MULTIPLYING BY 1 
DOESN'T CHANGE ANYTHING.
SO A LOT OF TIMES YOU'LL SEE 
TEXTBOOK JUST LEAVE THIS OFF.
SO TO SOLVE FOR T, 
WE JUST DIVIDE BY 0.08,
WHICH REMEMBER IS JUST 
THE CONTINUOUS INTEREST RATE.
SO THIS SIMPLIFIES TO JUST T,
AND THEN WE'LL GET A DECIMAL 
APPROXIMATION OF THIS QUOTIENT
ON THE CALCULATOR.
SO IT'LL TAKE APPROXIMATELY, 
LET'S SAY, 8.7 YEARS.
SO THERE'S A SHORTCUT 
TO FINDING DOUBLING TIME
WHEN TALKING 
ABOUT CONTINUOUS INTEREST
OR EXPONENTIAL GROWTH.
THIS SHORTCUT 
IS THE DOUBLING TIME,
WILL ALWAYS BE EQUAL 
TO NATURAL LOG 2
DIVIDED BY THE ANNUAL INTEREST 
RATE R,
OR FOR EXPONENTIAL GROWTH
IT WOULD BE THE EXPONENTIAL 
GROWTH RATE K.
BUT I'M NOT A BIG FAN 
OF MEMORIZING THIS FORMULA
BECAUSE WE CAN ALWAYS 
WORK IT OUT LIKE THIS
SHOWING THAT WE DO UNDERSTAND 
EXACTLY WHAT'S HAPPENING
IN THIS SITUATION.
OKAY. I HOPE YOU FOUND 
THIS HELPFUL.
