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SECTION 4.4. LOGARITHMIC FUNCTIONS. LOGARITHMIC FUNCTIONS. The logarithm (base b) of a number is the power to which b must be raised to get that number. EXAMPLES:. (a) log 4 16 =. 2. (e) log 5 1 =. 0. (b) log 3 27 =. 3. (f) log 2 1/8 =. - 3. (g) log 10 .1 =. - 1.

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SECTION 4.4

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Section 4 4

SECTION 4.4

  • LOGARITHMIC FUNCTIONS


Logarithmic functions

LOGARITHMIC FUNCTIONS

The logarithm (base b) of a number is the power to which b must be raised to get that number.


Examples

EXAMPLES:

(a) log 4 16 =

2

(e) log 5 1 =

0

(b) log 3 27 =

3

(f) log 2 1/8 =

- 3

(g) log10 .1 =

- 1

(c) log e e 4 =

4

(d) log 2 32 =

5

(h) log 9 27 =

3/2


Logarithmic functions1

LOGARITHMIC FUNCTIONS

Recall that only one-to-one functions have an inverse.

Exponential functions are one-to-one.

Their inverses are logarithmic functions.


Logarithmic functions2

LOGARITHMIC FUNCTIONS

Example:

Change the exponential expressions to logarithmic expressions.

1.23 = m

eb = 9

a4 = 24


Logarithmic functions3

LOGARITHMIC FUNCTIONS

Example:

Change the logarithmic expressions to exponential expressions.

loga4 = 5

loge b = - 3

log3 5 = c


Domain of a logarithmic function

DOMAIN OF A LOGARITHMIC FUNCTION

Since the logarithmic function is the inverse of the exponential, the domain of a logarithmic is the same as the range of the exponential.


Domain of a logarithmic function1

DOMAIN OF A LOGARITHMIC FUNCTION

Example:

Find the domain of the functions below:

F(x) = log2 (1 - x)


Special logarithms

SPECIAL LOGARITHMS

Logarithm to the base 10.

Ex:log 100 = 2

COMMON

LOGARITHM

Logarithm to the base e.

Ex:ln e 2 = 2

NATURAL

LOGARITHM


The natural logarithmic function

THE NATURAL LOGARITHMIC FUNCTION

  • Graph the function g(x) = lnx in the same coordinate plane with f(x) = ex

  • Notice the symmetry with respect to the line y = x.


F x e x g x lnx

f(x) = exg(x) = lnx

Compose the two functions:

g(f(x)) = ln ex = x f(g(x)) = eln x = x

We can see graphically as well as algebraically that these two functions are inverses of each other.


Section 4 4

Given f(x) = bx

Then f -1(x) = log b x


Graphs of logarmithmic functions

GRAPHS OF LOGARMITHMIC FUNCTIONS

1.The x-intercept is 1.

2.The y-axis is a vertical asymptote of the graph.

3.A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1.

4.The graph is continuous.


Graphing logarithmic functions using transformations

GRAPHING LOGARITHMIC FUNCTIONS USING TRANSFORMATIONS

Graph f(x) = 3log(x – 1). Determine the domain, range, and vertical asymptote of f.


Example

EXAMPLE

Graph the function f(x) = ln(1 - x).

Determine the domain, range, and vertical asymptote.


Solving a logarithmic equation

SOLVING A LOGARITHMIC EQUATION

Solve: log3(4x – 7) = 2

Solve: logx64 = 2


Using logarithms to solve exponential equations

USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS

Solve: e2x = 5


Example1

EXAMPLE

DO EXAMPLE 10 ON ALCOHOL AND DRIVING


Section 4 4

  • CONCLUSION OF SECTION 4.4


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