8.4 Logarithmic Functions

Objectives:1. Write logarithmic function in exponential

form and back2. Evaluate logs with and without calculator3. Evaluate the logarithmic function4. Understand logs and inverses5. Graph logarithmic function

Vocabulary:logarithm, common logarithm, natural

logarithm

In last section, 8.3, we have learned that the

Interest problem that if the interest of a bank

account of initial asset P is 5% compounded

and the total asset after t years is described by

the model (exponential growth):

Yearly: At= P (1 + 0.05 )t

Monthly: At= P (1 + 0.05 / 12)12·t

Daily: At= P (1 + 0.05 / 365)365·t

Continuously: At= P e 0.05·t

In each case, as long as we know the duration

of the asset deposited in the bank, the t, we

can calculate the final (total) asset:

Yearly: A5= 1200 (1 + 0.05 )5

Monthly: A10= 3500 (1 + 0.05 / 12)12·10

Daily: A2= 9560 (1 + 0.05 / 365)365·2

Continuously: A6= 27890 e 0.05·6

Now we would like to ask a reverse question: How long does the initial deposit (investment) take to reach the target asset value?

4 to what power gives me 64?

3 to what power gives me 81?

2 to what power gives me .125?

1/4 to what power gives 256?

32 to what power gives me 2?

Definition Logarithm of y with base b

Let b and y be positive numbers, and b ≠ 1. logby = x if and only if y = b x

Definition Exponential FunctionThe function is of the form: f (x) = a bx, where a ≠

0, b > 0 and b ≠ 1, x R.

Simple Exponential Function

Let b and y be positive numbers, and b ≠ 1.

y = b x

Example 1a) log39

b) log41

c) log5(1/25)

Special Logarithm Values

a) logb1 = 0

b) logbb = 1

Note: Some other special logarithm values are:

c) logb0 = undefined

d) logb (-2) = undefined

Challenge QuestionOne student said since (-3)2 = 9, so, log(-3)9

= 2, why do we need to constrain the base to be positive in the definition?

Answer to this Question

If the negative number can be used as a base, when we are going to discuss the more general situation such as log-33, this will turn out

3 = (-3)x. And this will never be true for any real

number x.

Practice

a) log13169

b) log1001

c) log25252

d) log25255

Example 2 Evaluate the expression

a) log464

b) log20.125

c) log1/4256

d) log322

Practice Evaluate the expression

a) log327

b) log40.0625

c) log1/16256

d) log642

e) If k > 0 and k ≠ 1, logk1

Definition Common Logarithm

log10x = logx

Definition Natural Logarithm

logex = lnx

Example 3 Evaluate the common and natural logarithm

a) log4

b) ln(1/5)

c) lne-3

d) log(1/1000)

Practice Evaluate the common and natural logarithm

a) log7

b) ln0.25

c) log3.8

d) ln3

e) lne2007

The logarithmic function with base b is defined as

g(x) = logbxwith domain x R+.

From the definition of a logarithm, we noticed that the logarithmic function

g(x) = logbx is the inverse of the exponential function

f(x) = bx

Because

This means that they offset each other, or they are “undo” each other.

xbf(g(x))xblog

xblogg(f(x)) xb

Example 4 The Richter scale is used for measuring the magnitude of an earthquake. The Richter magnitude R is given by the model

R = 0.67 log(0.37E) + 1.46

Where E is the energy (in kilowatt-hours)

released by the earthquake.

a) Suppose an earthquake releases 15,500, 000,000 million kwh of energy. What is the earthquake’s magnitude? (7.998)

b) How many kwh of energy would the earthquake above have to release in order to increase its magnitude by one-half of a unit on the Richter scale? (8.6417E10)

Example 5 Evaluate the common and natural logarithm

a) 10log4

b) eln(1/5)

c) log5125 x

d) lne – 3 x

Practice Simply

a) 10log5x

b) log100002x

From the definition of a logarithm, we noticed that the logarithmic function

g(x) = logbx is the inverse of the exponential function

f(x) = bx

Because

This means that they offset each other, or they are “undo” each other. These two functions are inverse to each other.

xbf(g(x))xblog

xblogg(f(x)) xb

The graph of the logarithmic functionf(x) = bx ( b > 1)

is

x-axis andy-axis arehorizontaland verticalasymptotes.

the graph of its inverse function

g(x) = logbxTwo graphs are symmetry to the line y = x

The graph of the logarithmic functionf(x) = bx ( 0 < b < 1)

is

x-axis andy-axis arehorizontaland verticalasymptotes.

the graph of its inverse function

g(x) = logbxTwo graphs are symmetry to the line y = x

Example 6 Find the inverse of the function

a) y = log8x

b) y = ln(x – 3)

Answera) y = 8x

b) y = ex + 3

Practice Find the inverse of

a) y = log2/5x

b) y = ln(2x – 10)

Answera) y = (2/5)x

b) y = (ex + 10)/2

Function FamilyThe graph of the function

y = f(x – h) k x – h = 0, x = h

is the graph of the functiony = f(x)

shift h unit to the right and k unit up/down.

The graph of the functiony = f(x + h) k x + h = 0,

x = –h is the graph of the function

y = f(x)shift h unit to the left and k unit up/down.

Logarithmic Function FamilyThe graph of the logarithmic function

y = logb(x h) k

has1) The domain is x > –h (x > h).2) The line x = –h (x = h) is the vertical

asymptote.3) If 0 < b < 1, the curve goes down.

If b > 1, the curve goes up.4) The function graph is shifted h units

horizontally and k units vertically from the graph y = logbx

Example 7 Graph the function, state domain and range.

a) y = log1/2 (x + 4) + 2 b) y = log3(x – 2) – 1

1- 4

1 2

0

0

Assignment:8.4 P490 #17-76 even - Show work

8.4 Logarithmic Functions