Logarithmic Functions and Their Graphs

Review: Changing Between Logarithmic and Exponential Form

• If x > 0 and 0 < b ≠ 1, then

if and only if .

• This statement says that a logarithm is an exponent. Because logarithms are exponents, we can evaluate simple logarithmic expressions using our understanding of exponents.

log ( )by x yb x

Evaluating Logarithmsa.) because .

b.) because .

c.) because

d.) because

e.) because

2log 8 3 32 8

3log 3 1 2 1 23 3

5

1log 2

25 2

2

1 15 .

5 25

4log 1 0 04 1.

7log 7 1 17 7.

Basic Properties of Logarithms• For 0 < b ≠ 1 , x > 0, and any real number y,

logb1 = 0 because b0 = 1.

logbb = 1 because b1 = b.

logbby = y because by = by.

becauselogb xb x

Evaluating Logarithmic and Exponential Expressions

(a)

(b)

(c)

2log 8 32log 2 3.

3log 3 1 23log 3 1 2.

6log 116 11.

Common Logarithms – Base 10• Logarithms with base 10 are called common

logarithms.• Often drop the subscript of 10 for the base when

using common logarithms.• The common logarithmic function:

y = log x if and only if 10y = x.

Basic Properties of Common Logarithms• Let x and y be real numbers with x > 0.

log 1 = 0 because 100 = 1.

log 10 = 1 because 101 = 10.

log 10y = y because 10y = 10y .

because log x = log x.

• Using the definition of common logarithm or these basic properties, we can evaluate expressions involving a base of 10.

log10 x x

Evaluating Logarithmic and Exponential Expressions – Base 10

(a)

(b)

(c)

(d)

log100 10log 100 2

5log 10 1 5log10 1

5

1log1000 3

1log10

3log10 3log610 6

Solving Simple Logarithmic Equations• Solve each equation by changing it to exponential

form:a.) log x = 3 b.)

a.) Changing to exponential form, x = 10³ = 1000.

b.) Changing to exponential form, x = 25 = 32.

2log 5x

Natural Logarithms – Base e

• Logarithms with base e are natural logarithms.– We use the abbreviation “ln” (without a subscript)

to denote a natural logarithm.

Basic Properties of Natural Logarithms• Let x and y be real numbers with x > 0.

ln 1 = 0 because e0 = 1.

ln e = 1 because e1 = e.

ln ey = y because ey = ey.

eln x = x because ln x = ln x.

Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y

= log x into the graph of the given function.a.) g(x) = ln (x + 2)

The graph is obtained by translating the graph of y = ln (x) two units to the LEFT.

Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y

= log x into the graph of the given function.c.) g(x) = 3 log x

The graph is obtained by vertically stretching the graph of f(x) = log x by a factor of 3.

Transforming Logarithmic Graphs• Describe how to transform the graph of y = ln x or y

= log x into the graph of the given function.d.) h(x) = 1+ log x

The graph is obtained by a translation 1 unit up.