Logarithmic functions their graphs
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Logarithmic Functions & Their Graphs. Section 3.2. Log Functions & Their Graphs. In the previous section, we worked with exponential functions. What did the graph of these functions look like?. Log Functions & Their Graphs. Earlier in the year, we covered “inverse functions”

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Logarithmic Functions & Their Graphs

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Logarithmic functions their graphs

Logarithmic Functions & Their Graphs

Section 3.2


Log functions their graphs

Log Functions & Their Graphs

In the previous section, we worked with exponential functions.

What did the graph of these functions look like?


Log functions their graphs1

Log Functions & Their Graphs

Earlier in the year, we covered “inverse functions”

Do exponential functions have an inverse?

By looking at the graphs of exponential functions, we notice that every graph passes the horizontal line test.

Therefore, all exponential functions have an inverse


Log functions their graphs2

Log Functions & Their Graphs

The inverse of an exponential function with base a is called the logarithmic function with base a

For x > 0, a > 0 and a ≠ 1


Log functions their graphs3

Log Functions & Their Graphs

  • In other words:

    really means that a raised to the power of y is equal to x

  • The log button on your calculator refers to the Log base 10

    • This is referred to as the Common Logarithm


Log functions their graphs4

Log Functions & Their Graphs

Another common logarithm is the Log base e

This is referred to as the Natural Logarithmic Function

This function is denoted:


Log functions their graphs5

Log Functions & Their Graphs

Write the following logarithms in exponential form.


Log functions their graphs6

Log Functions & Their Graphs

Write the exponential equations in log form


Log functions their graphs7

Log Functions & Their Graphs

Evaluate the following logarithms:

Since a raised to the

power of zero is equal to 1,

Since a raised to the

power of one is equal to a

= 0

= 1


Log functions their graphs8

Log Functions & Their Graphs

Now that we know the definition of a logarithmic function, we can start to evaluate basic logarithms.

What is this question asking?

2 raised to what power equals 8?

2³= 8

x = 3


Log functions their graphs9

Log Functions & Their Graphs

Evaluate the following logarithms:


Log functions their graphs10

Log Functions & Their Graphs

Properties of Logarithms


Log functions their graphs11

Log Functions & Their Graphs

Using these properties, we can simplify different logarithmic functions.

= x

From our third property, we can evaluate this log function to be equal to x.


Log functions their graphs12

Log Functions & Their Graphs

Use the properties of logarithms to evaluate or simplify the following expressions.


Log functions their graphs13

Log Functions & Their Graphs

In conclusion, what does the following statement mean?

“10 raised to the power of y is equal to z”


Logarithmic functions their graphs1

Logarithmic Functions & Their Graphs

Section 3.2


Log functions their graphs14

Log Functions & Their Graphs

Yesterday, we went over the basic definition of logarithms.

Remember, they are truly defined as the inverse of an exponential function.


Graphs of log functions

Graphs of Log Functions

Fill in the following table and sketch the graph of the function f(x) for:

f(x) =


Graphs of log functions1

Graphs of Log Functions

Remember that the function is actually the inverse of the exponential function

To graph inverses, switch the x and y values

This is a reflection across the line y = x


Graphs of log functions2

Graphs of Log Functions

Fill in the following table and sketch the graph of the function f(x) for:


Graphs of log functions3

Graphs of Log Functions

The nature of this curve is typical of the curves of logarithmic functions.

They have one x-intercept and one vertical asymptote

Reflection of the exponential curve across the line y = x


Graphs of log functions4

Graphs of Log Functions

Basic characteristics of the log curves

Domain: (0, ∞)

Range: (- ∞, ∞)

x-intercept at (1, 0)

Increasing

1-1 → the function has an inverse

y-axis is a vertical asymptote

Continuous


Graphs of log functions5

Graphs of Log Functions

Much like we had shifts in exponential curves, the log curves have shifts and reflections as well

Graphing will shift the curve 1 unit to the right

Graphing will shift the curve vertically up 2 units


Graphs of log functions6

Graphs of Log Functions

Much like we had shifts in exponential curves, the log curves have shifts and reflections as well

Graphing will reflect the curve over the vertical asymptote

Graphing will reflect the curve over the x-axis


Graphs of log functions7

Graphs of Log Functions

Sketch a graph of the following functions.


Graphs of log functions8

Graphs of Log Functions

Domain:

(3, ∞)

x-intercept:

(4, 0)

Asymptote:

x = 3


Graphs of log functions9

Graphs of Log Functions

Domain:

(1, ∞)

x-intercept:

( , 0)

Asymptote:

x = 1


Graphs of log functions10

Graphs of Log Functions

Domain:

(- ∞, 3)

x-intercept:

(2, 0)

Asymptote:

x = 3


Graphs of log functions11

Graphs of Log Functions

Notice that the first piece of information we have been gathering on the graphs is the domain.

For x > 0, a > 0 and a ≠ 1

This means that whatever value is in the place of x must be positive


Graphs of log functions12

Graphs of Log Functions

What would the domain of this function be?

(0, ∞)

→ - x > 0

→ x < 0

→ The domain would be: (-∞, 0)


Graphs of log functions13

Graphs of Log Functions

Find the domain of the following logarithms.

a)

b)

c)


Applications

Applications

The model below approximates the length of a home mortgage of $150,000 at 8% interest in terms of the monthly payment. In the model, t is the number of years of the mortgage and x is the monthly payment in dollars.


Applications1

Applications

Use this model to approximate the length of a mortgage if

the monthly payment is $1,300.

By putting $1,300 in for x, you should get a time of 18.4

years


Applications2

Applications

  • How much would you end up paying in interest using this same example?

  • Paying $1,300 a month for 18.4 years

    → Pay a total of (18.4) (1,300) =

  • Therefore, interest would be equal to $137,040.

$287,040


Applications3

Applications

Using this same model, approximate the length of a mortgage when the monthly payment is:

a) $1,100.65andb) $1,254.68


Applications4

Applications

a) $1,100.65andb) $1,254.68

What would the difference in amount paid be for each of

these mortgages?

30 years

20 years


Applications5

Applications

A principal P, invested at 6% interest compounded continuously, increases to an amount K times the original principal after t years, where t is given by:

How long will it take the original investment to double?

By putting in 2 for K, we get t = 11.55 years


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