Logarithmic Functions• Recall that for a > 0, the exponential function f(x) = ax

is one-to-one. This means that the inverse function exists, and we call it the logarithmic function with base a, written loga.

• We have:

• Example. log10 1000 = 3, log10 = 0.5, log10 0.01 = –2.

• The notation log x is shorthand for log10 x, and log x is called the common logarithm.

• The notation ln x is shorthand for loge x, and ln x is called the natural logarithm.

,1a

. log y

a axxy

10

Graphs of 10x and log x

Note that the domain of log x is the set of positive x values.

Logarithmic Identities and Properties

• From the definition of the logarithm it follows that

• From the fact that the exponential and logarithmic functions are inverse functions it follows that

• Since loga x is one-to-one, it follows that

• Since the graphs of loga and logb intersect only at x = 1,

.01log

1log

a

a a

.log

log

xa

xax

a

xa

. loglog vuvu aa

.or 1 loglog bauuu ba

Fundamental Properties of Logarithms

• The following three properties of logarithms can be proved by using equivalent exponential forms.

• Problems. log 2 + log 5 = ???, log 250 – log 25 = ???, log 101/3 = ???

number. real a ,loglog

logloglog

logloglog

nxnx

yxy

x

yxxy

an

a

aaa

aaa

Write the expression as a single logarithm

2

2

21

)1(

1log

)1(log)1(log

)1(log2)1(log

21

x

x

xx

xx

a

aa

aa

Solving an equation using logarithms

• If interest is compounded continuously, at what annual rate will a principal of $100 triple in 20 years?

5.5%or 055.020

3ln

203ln

ln3ln

3

100300

20

20

20

r

r

e

e

e

PeA

r

r

r

rt

Change of Base Formula

• Sometimes it is necessary to convert a logarithm to base a to a logarithm to base b. The following formula is used:

• Compute log2 27 using common logs and your calculator.

• Check your answer:

.1,1,0,0,log

loglog baba

b

xx

a

ab

7549.430103.0

43136.1

2log

27log27log2

272 7549.4

Exponential Equations

• When solving an exponential equation, consider taking logarithms of both sides of the equation.

• Example. Solve 32x–1 = 17.

7895.13log

17log15.0

3log

17log12

17log3log)12(

17log3log 12

x

x

x

x

Solving an Exponential Equation for Continuous Compounding • Problem. A trust fund invests $8000 at an annual rate of 8%

compounded continuously. How long does it take for the initial investment to grow to $12,000? Solution. We must solve for t in the following equation.

years. 07.50.08

1.5 ln

1.5 ln08.0

5.18000

12000

120008000

08.0

08.0

t

t

e

e

t

t

Logarithmic Equations

• When solving a logarithmic equation, consider forming a single logarithm on one side of the equation, and then converting this equation to the equivalent exponential form.

• Be sure to check any "solutions" in the original equation since some of them may be extraneous.

• Problem. Solve for x.

.4or 2

0)4)(2(

082x

form lexponentia equivalent ,2)2(

3)2(log

3)2(loglog

2

3

2

22

xx

xx

x

xx

xx

xx

Summary of Exponential and Logarithmic Functions; We discussed

• Definition of logarithm as inverse of exponential

• The common logarithm

• The natural logarithm

• Graphs of y = log x and y = ax

• Domain of the logarithm

• Fundamental properties of logarithms log of product log of quotient log of xn

• Change of base formula

• Solving exponential equations

• Solving logarithmic equations