Exponential Functions
What is the general form of an exponential function?
f(x)=a⋅b^x
Convert 10x=1000 into logarithmic form.
x= log 1000
What is the formula for compound interest?
A=P(1+n/r) nt
What is the formula for the future value of an annuity?
Answer: FV=P(1+r/n)nt−1/ r/n
If a bacteria culture starts with 100 cells and doubles every 4 hours, how many will there be in 12 hours?
800
Evaluate log232
5
If $500 is invested at 6% compounded monthly for 3 years, find the final amount.
Answer: 500(1+0.06/12)12x3≈598.30
Explanation: Use the formula with n=12, r=0.06 , and t=3
Suppose Adam wants to have $750,000 in his IRA at the end of 30 years. He decides to invest in an annuity paying 6% interest, compounded annually. What does he have to contribute each year to reach this goal? Enter your answer, rounded to the nearest cent. Omit the dollar sign and the comma.
d=$9,486.68
Begin by using the savings annuity formula PN=d((1+r/k)Nk−1)/ r/k and solve for d. From the question, P30=$750,000, r=0.06 and k=1 compounding period per year. Substitute these values into the formula, and solve for d, the yearly contribution.$750,000=d((1+0.06/1)30×1−1)/ 0.06/1
$750,000=d (1.0630−1)/0.06 $750,000=79.0581862d
d=$9,486.68
The population of a city follows P(t)=50,000(1.02)t. How long will it take for the population to reach 100,000?
About 35 years
Solve for x: log5(x)=3
x = 53= 125
What is the difference between simple and compound interest?
Compound interest includes interest on prior interest, while simple interest does not.
Suppose Laura invests in an annuity to have $10,000 after 15 years. If the annuity pays 2% annual interest, compounded quarterly, how much will she need to invest each quarter to meet her goal? Enter your answer, rounded to the nearest dollar.
143
Use the formulaPN=d((1+r/k)Nk−1)/r/k and solve for d. The question states that P15=$10,000, r=0.02, and k=4 compounding periods per year. Substitute these values into the formula, simplify and solve.$10,000=d((1+0.02/4)15×4−1)/0.02/4
10,000=d(1.00560−1)/0.005
$10,000=69.770030d
d=$143.33
Round the answer to the nearest dollar: 143.
Find the intersection points of f(x)=3x and g(x)=2x+1
x≈0.63
If logb64=3, what is b?
b=4
b3=64⇒b=4.
How often is interest compounded in continuous compounding?
Infinitely
Suppose you find an annuity that pays 8% annual interest, compounded annually. If you invest in this annuity and contribute $10,000 annually for 10 years, how much money will be in the annuity after 10 years? Enter your answer rounded to the nearest hundred dollars
144900
The question asks us to find the value of A(10). We will use the savings annuity formula
A(t)=d⋅((1+r/n)n⋅t−1)/ r/n
The question tells us that r=0.08, d=$10,000, n=1 compounding periods per year, and t=10 years. Substitute these values into the formula gives
A(10)=10,000⋅((1+0.08/1)1⋅10−1)/ 0.08/1
A(10)=$144,865.62.
Our final answer is 144900.
Solve for x: 5x+1=2⋅5x+75
x=2
Solve log(x)+log(x−3)=1
Use the logarithm product rule: log(x(x−3))=1.
Rewrite in exponential form: x(x−3)=101
Solve quadratic: x2−3x−10=0
Factor: (x−5)(x+2)=0⇒x=5,−2
Logarithm domain restriction: x>3, so discard x=−2
Final Answer: x=5
If $1,000 is invested at 5% continuously, what is the value after 4 years? What formula is used for this question?
Answer: A=1000e0.05×4≈1220.34
Explanation: Use A=Pert
Tom wishes to purchase a property that's been valued at $300,000. He has 25% of this amount available as a cash deposit, and will require a mortgage for the remaining amount. The bank offers him a 25-year mortgage at 2% interest with monthly payments. Calculate the total interest Tom will pay. Give your answer in dollars to the nearest ten dollars.
61100
First, we note that Tom requires a mortgage on $300,000×75%=$225,000. To calculate the monthly repayments we must apply the formula for P0 and solve for d, that is,
P0=d(1−(1+r/k)−Nk/(r/k).We have P0=$225,000, r=0.02, k=12, N=25, so substituting in the numbers into the formula gives
$225,000=d(1−(1+0.02/12)−25⋅12)/ (0.0212)
,that is,
$225,000=235.9301d ⟹ d=$953.67.
So the total interest will be
I=$953.67 × 25 × 12−$225,000=$61,100.
to the nearest $10.