# 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.

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