Logarithmic Functions and Their Graphs

Consider xf x a

This is a one-to-one function, therefore it has an inverse.

The inverse is called a logarithm function.

Example:416 2 24 log 16 Two raised to what power

is 16?

The most commonly used bases for logs are 10: 10log logx x

and e: log lne x x

lny x is called the natural log function.

logy x is called the common log function.

Definition of Logarithmic Function

b > 0; b 1

Logarithmic Form Exponential Form

y = logb x x = by

The log to the base “b” of “x” is the exponent to which “b” must

be raised to obtain “x”

y = log10 x

y = log e x

x = 10 y

x = e y

Change from Logarithmic To Exponential Form

Log 2 8 = 3 8 = 23

5 = 25 ½Log 25 5 = ½

Change from Exponential To Form Logarithmic

49 = 7 2 log 7 49 = 2

1/5 = 5 –1 log 5 (1/5) = -1

Properties of Logarithmic FunctionsIf b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Log15 1 = 0

Log10 10 = 1

Log5 5x = x

3log x = x 3

150 = 1

101 = 10

5x = 5x

The Decibel Scale

The decibel level D of a sound of intensity I , measured in watts per

square meter (W/ m2) is given by  

 

where I0 = 10–12 W/ m2 is the intensity of the least audible sound that

an average healthy person can hear. 

Sound Intensity, W/ m2 Sound

  1.0 10–12 Threshold of hearing

5.2 10–10 Whisper

3.2 10–6 Normal conversation

8.5 10–4 Heavy traffic

3.2 10–3 Jackhammer

1.0 100 Threshold of pain

8.3 102 Jet plane with afterburner

0

log10DI

I

The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by

where E0 = 104.40 joules is the energy released by a small reference

earthquake.

0EE

log32

M

Magnitude on Richter scale Destructive power

M < 4.5 Small4.5 < M < 5.5 Moderate5.5 < M < 6.5 Large6.5 < M < 7.5 Major7.5 < M Greatest

The Richter Scale

Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…

f

x y = 2 x

–31

8

–21

4

–11

2

0 1

1 2

2 4

3 8

f –1

x = 2 y

1

8 –3 1

4 –2 1

2 –1

1 0

2 1

4 2

8 3

Ordered pairs reversed

y

x

y

5 10 –5

5

10

–5

f -1

x = 2y

or y = log2x

f y = 2x

y = x

DOMAIN of = (– , ) = RANGE of

RANGE of f = (0, ) = DOMAIN of

Logarithmic Function with Base 2

f

f -1

f -1