Why the Power Rule/Change of Base Rule

LetThenSo raising to the rth power yields Therefore

“the log of x to the rth power is r times the log of x”

Again

Or

logby xyx b

rr y r yx b b

log logrb bx r y r x

loglog loglog logb ab n n nb b aa

log

loglog n

nb

a

ba log log logb a

b by b iff y a

exponent

Solving Exponential Equations

1. Solve: done!2. Given 50 grams of a radioactive substance with a

half life of 15 days, how long until only 20 grams remain?

3. Given an initial population of 7000 and a growth rate of 6% per year, when will the population reach 10,000?

Common Logs versus Natural Logs

1.042 1.04 log 2t t

10logy x log lney x x

Solving Exponential Equations Two Ways

1. Solve two ways – first by taking the ln( ) of both sides and using log rules to simplify; second by rewriting it as and converting it to a logarithmic equation.

2. If Radium has a half life of 1690 years, how old is an object if the remaining Radium is down to 30%

3. If the population of Columbus OH was 632,910 in 1990 and 711,265 in 2000, find the exponential equation which models its population and determine when the population will the reach 1 million.

4000 2000 1 0.04t

xy b

Interest compounded k times per year

If interest is compounded k times per year – divide the interest rate by k and multiply the number of years by k.

Example: If you invest $1000 at 8% interest, how much do you earn after 5 years if interest is compounded

Quarterly? Monthly? Weekly? Daily?How long to earn $2000 if interest is compounded

Quarterly? Monthly? Weekly? Daily?

1k t

rFV PV

k

Present Value

Future Value

Interest Compounded Continuously

If we increase k, the number of times integers is compounded per year, we can show …

If you invest $1000 at 8% interest, how much do you earn after 5 years if interest is compounded

Continuously? How long to earn $2000 if interest is compounded

Continuously?

lim 1k n

r t

k

rPV P e

kV

e and compounding continuously

Recall: The limit exists and defines the value of

e which is approximately 2.718281828459… If interest is compounded continuously it can be shown that

To evaluate let so

lim1

1n

ne

n

lim 1 rt

tk

kPV P e

k

rV

lim 1kt

k

rPV

k

lim lim1 1

1 1

rt rn n t

r t

n ne

n nPV PV PV

kn

r n iff k

APY and Present Value

A common comparison for comparing investments, Annual Percentage Yield (APY) is the percentage rate, r, that compounded annually would yield the same return as a given interest rate,r0, with a given compounding period

This simplifies to

Find the APV for $1000 at 8% interest compounded Quarterly! Monthly! Weekly! Daily! Continuously !

01 1k t

t rPV PV

kr

001 1 1 rk

or ek

rr

r continuous compounding case

Present Value?

How much do you have to invest now at 8% interest compounded quarterly (monthly, weekly, daily, continuously) so that in 5 years you will have $2000?

1k t

rFV PV

k

50.08

2000 1k

PVk

50.08

2000 1k

PVk

1k t

r trPV FV or PV FV e

k