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Logarithmic Functions & Their Graphs

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Logarithmic Functions & Their Graphs

Section 3.2

In the previous section, we worked with exponential functions.

What did the graph of these functions look like?

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

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

- 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

Another common logarithm is the Log base e

This is referred to as the Natural Logarithmic Function

This function is denoted:

Write the following logarithms in exponential form.

Write the exponential equations in log form

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

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

Evaluate the following logarithms:

Properties of Logarithms

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.

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

In conclusion, what does the following statement mean?

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

Logarithmic Functions & Their Graphs

Section 3.2

Yesterday, we went over the basic definition of logarithms.

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

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

f(x) =

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

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

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

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

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

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

Sketch a graph of the following functions.

Domain:

(3, ∞)

x-intercept:

(4, 0)

Asymptote:

x = 3

Domain:

(1, ∞)

x-intercept:

( , 0)

Asymptote:

x = 1

Domain:

(- ∞, 3)

x-intercept:

(2, 0)

Asymptote:

x = 3

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

What would the domain of this function be?

(0, ∞)

→ - x > 0

→ x < 0

→ The domain would be: (-∞, 0)

Find the domain of the following logarithms.

a)

b)

c)

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.

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

- 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

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

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

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

What would the difference in amount paid be for each of

these mortgages?

30 years

20 years

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