Section 6.4Solving Logarithmic and Exponential Equations

• Suppose you have $100 in an account paying 5% compounded annually.– Create an equation for the balance B after t years– When will the account be worth $200?

• In the previous example we needed to solve for the input

• Since exponential functions are 1-1, they have an inverse

• The inverse of an exponential function is called the logarithmic function or log

• In other words

• If b = 10 we have the common log

If log then yby x b x

If log then 10 yy x x

Example• Rewrite the following expressions using logs

4

5

3

3

10 10,000

10 0.00001

10 3.16227766

5 125

8 2

• Logarithms are just exponents• Evaluate the following without a calculator by

rewriting as an exponential equation:

log1

log10

log(100,000)

log( 10)

log( 10)

y

y

y

y

y

• Logarithms are inverses of exponential functions so

• To see why this works rewrite the logarithm as an exponential equation

• Also• To see why this works rewrite the exponential

equation as a logarithmic equation• Evaluate

log ( )mb bb m

27 3log 2.222 log ( 1)7.5

5log(10 ) 7 3 log 25x

log ( )mb b m

Properties of the Logarithmic Function

log

If log then

log 1 0 and log 1

log ( ) for all

for 0

log ( ) log log

log log log

log ( ) log

b

yb

b b

xb

x

b b b

b b b

tb b

y x b x

b

b x x

b x x

xy x y

xx y

y

x t x

Now the log we have in our calculator is the common log so b = 10. There is also the natural log, ln on our calculator where b = e. It has all the same properties.

The Natural Logarithm

btb

bab

a

baab

xxe

xxe

e

xexy

t

x

x

y

ln)ln(

lnlnln

lnln)ln(

0for

allfor)ln(

1lnand01ln

thenlnIf

ln

Evaluate without a calculator210 100

10 22(0.87)

x

q

e

10

100log

10

1ln

e

Simplify without a calculator

Change of Base Formula• It turns out we can write a log of a base as a

ratio of logs of the same base• This is useful if our solution contains a log that

does not have a base of 10 or e• The Change of Base Formula is

• For us we typically use 10 or e for a since that is what we have in our calculator

log ( )log

log ( )a

ba

xx

b

• Reminder: The half-life of a substance is the amount of time it takes for a decreasing exponential function to decay to half of its initial value

• The half-life of iodine-123 is about 13 hours. You begin with 100 grams of iodine-123.– Write an equation that gives the amount of iodine

remaining after t hours• Hint: You need to find your rate using the half-life

information

– Determine the number of hours for your sample to decay to 10 grams

• Reminder: Doubling time is the amount of time it takes for an increasing exponential function to grow to twice its previous level

• What is the doubling time of an account that pays 4.5% compounded annually? Quarterly?

• Recall that the population of Phoenix went up by 45.3% between 1990-2000. Assuming that growth remained steady, what is the doubling time of the Phoenix population?

• Any exponential function can be written as Q = abt or Q = aekt

–Then b = ekt so k = lnb• Convert the function Q = 5(1.2)t into the form

Q = aekt

–What is the annual growth rate?–What is the continuous growth rate?

• Convert the function Q = 10(0.81)t into the form Q = aekt

–What is the annual decay rate?–What is the continuous decay rate?

• Let’s try a few from the chapter

6.4 – 1, 3, 5, 7, 9, 19, 27, 29, 37, 49