Ex 2: Compounded Interest with Logarithms


– YOU DEPOSIT $5,500 INTO A BANK THAT PAYS 3% ANNUAL INTEREST
COMPOUNDED MONTHLY. WHAT EQUATION CAN BE USED TO
DETERMINE THE ACCOUNT BALANCE AFTER T YEARS? WHAT IS THE ACCOUNT BALANCE
AFTER TWO YEARS? AND WHEN WILL THE BALANCE
REACH $6,000? SO WE’LL BE USING THE COMPOUNDED
INTEREST FORMULA GIVEN HERE BELOW WHERE P REPRESENTS THE PRINCIPLE
OR INITIAL INVESTMENT AMOUNT, R IS THE ANNUAL INTEREST RATE
EXPRESSED AS A DECIMAL. N IS THE NUMBER OF COMPOUNDS
PER YEAR. NOTICE HOW N OCCURS HERE
AND HERE IN THE EXPONENT. T IS THE TIME IN YEARS, AND “A”
IS THE AMOUNT AFTER T YEARS. SO TO FIND THE EQUATION
THAT MODELS THIS ACCOUNT BALANCE WILL HAVE “A”=THE PRINCIPLE, A STARTING AMOUNT OF $5,500
x THE QUANTITY 1 + THE ANNUAL INTEREST RATE
EXPRESSED AS A DECIMAL. 3% AS A DECIMAL WOULD BE 0.03. REMEMBER WE DROP THE PERCENT
SIGN AND THEN DIVIDE BY 100 OR MOVE THE DECIMAL POINT
TO THE LEFT TWO PLACES. N IS THE NUMBER OF COMPOUNDS
PER YEAR. IT’S COMPOUNDED MONTHLY, AND
THERE ARE 12 MONTHS IN A YEAR SO N IS 12. SO WE’RE GOING TO RAISE THIS
TO THE POWER OF N x T. SO, AGAIN, N IS 12 AND T
IS THE UNKNOWN TIME IN YEARS. LET’S GO AHEAD
AND FIND THIS SUM HERE. WE WOULD HAVE “A”=5,500 x 1
+ 0.03 DIVIDED BY 12. SO WE HAVE 1.0025
RAISED TO THE POWER OF 12T. AND NOW FOR THE SECOND QUESTION, TO FIND THE ACCOUNT BALANCE
AFTER TWO YEARS WE JUST SUBSTITUTE TWO FOR T. SO WE COULD SAY “A” OF 2=5,500
x 1.0025 RAISED TO THE POWER OF 12 x 2
WHICH OF COURSE WOULD BE 24. SO NOW WE’LL GO BACK
TO THE CALCULATOR AND WE’LL GO AHEAD AND ROUND
THIS TO THE NEAREST CENT.   SO THE ACCOUNT BALANCE
WOULD BE $5,839.66.   NOW, FOR THIS LAST QUESTION
WE WANT TO KNOW WHEN THE BALANCE WILL REACH
$6,000. SO THEY’RE GIVING US “A,” WE WANT TO SOLVE THE EQUATION
FOR T. SO NOW WE’LL BE SOLVING
AN EXPONENTIAL EQUATION IN WHICH WE’LL USE LOGARITHMS. SO “A” IS GOING TO BE 6,000. WE WANT TO SOLVE THIS EQUATION
FOR T. SO THE FIRST STEP IS TO ISOLATE
THIS EXPONENTIAL PART. SO WE’LL DIVIDE BOTH SIDES
BY 5,500. THIS SIMPLIFIES TO 1. 6,000 DIVIDED BY 5,500. NOTICE HOW THIS IS A REPEATING
DECIMAL, SO WE’LL GO AHEAD AND LEAVE THIS
AS A SIMPLIFIED FRACTION. SO I’LL PRESS MATH, ENTER,
ENTER. SO WE’LL LEAVE THE LEFT SIDE
OF THE EQUATION AS 12/11. SO OVER HERE WE’LL HAVE 1.0025
RAISED TO THE 12T POWER, AND NOW WE’RE GOING TO TAKE
THE NATURAL LOG OF BOTH SIDES OF THE EQUATION. WHEN WE DO THIS WE CAN APPLY
THE POWER OF PROPERTY OF LOGARITHMS HERE AND MOVE THIS EXPONENT OF 12T
TO THE FRONT. SO NOW WE HAVE THE EQUATION
NATURAL LOG OF 12/11=12T x NATURAL LOG OF 1.0025. NOW, WE’RE TRYING TO SOLVE
THIS EQUATION FOR T SO WE’RE GOING TO DIVIDE BOTH
SIDES OF THE EQUATION BY 12, AS WELL AS NATURAL LOG 1.0025.   NOTICE ON THE RIGHT SIDE 12/12
SIMPLIFIES TO 1, AS WELL AS THESE TWO
NATURAL LOGS. SO WE JUST HAVE T
ON THE RIGHT SIDE. SO T IS EQUAL TO THIS QUOTIENT
HERE. WE’LL HAVE TO GET A DECIMAL
APPROXIMATION ON THE CALCULATOR. SO THE NUMERATOR IS NATURAL LOG
12 DIVIDED BY 11 AND THE DENOMINATOR IS GOING
TO BE 12 NATURAL LOG 1.0025. NOTICE HOW WE HAVE
A SET OF PARENTHESIS AROUND THE NUMERATOR
AND DENOMINATOR TO MAKE SURE IT CALCULATES
THIS QUOTIENT CORRECTLY. SO T IS GOING TO BE
APPROXIMATELY, LET’S SAY 2.9 YEARS. COMPARING THIS TO OUR ANSWER
FOR THE SECOND QUESTION, THIS DOES SEEM
LIKE A REASONABLE ANSWER. OKAY. I HOPE YOU FOUND
THIS HELPFUL.  

6 Comments

  • Thank you so much! Stupid question – why do you use ln (natural log) instead of log? I can't figure it out. I know you use it to cancel out e, since its log to the base of e, but why do you use it in this case?

  • How do u find the interest rate of annuity…???

  • Thanks a lot. I was struggling on my homework and needed help.

  • Your videos are great, but on the Botton is very difficult to see when you write the examples. The letters are huge;-(

  • Super helpful! Thanks

  • ln(12/11)/ln(1.03)=t

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