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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) log10 .1 =

- 1

(c) log e e 4 =

4

(d) log 2 32 =

5

(h) log 9 27 =

3/2

LOGARITHMIC FUNCTIONS

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

Exponential functions are one-to-one.

Their inverses are logarithmic functions.

LOGARITHMIC FUNCTIONS

Example:

Change the exponential expressions to logarithmic expressions.

1.23 = m

eb = 9

a4 = 24

LOGARITHMIC FUNCTIONS

Example:

Change the logarithmic expressions to exponential expressions.

loga4 = 5

loge b = - 3

log3 5 = c

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 FUNCTION

Example:

Find the domain of the functions below:

F(x) = log2 (1 - x)

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

- 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) = ex g(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.

Then f -1(x) = log b x

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

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

USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS

Solve: e2x = 5

EXAMPLE

DO EXAMPLE 10 ON ALCOHOL AND DRIVING

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