Logarithmic Functions

x = 2y is an exponential equation.

If we solve for y it is called a logarithmic equation.

Let’s look at the parts of each type of equation:

Exponential Equationx = ay

exponent

base

number

/logarithm

y = loga xLogarithmic Equation

In General, a logarithm is the exponent to which the base must be Raised to get the number that you are taking the logarithm of.

Example: Rewrite in exponential form and

solve loga64 = 2

a2 = 64

a = 8

Example: Solve log5 x = 3

Rewrite in exponential form:

53 = x

x = 125

base number exponent

Example: Solve

7y = 1 49

y = –2

log7

1

49y

An equation in the form y = logb x where b > 0 and b ≠ 1 is called a logarithmic function.

Logarithmic and exponential functions are inverses of each other

logb bx = x

blogb x = x

Examples. Evaluate each:a. log8 8

4

b. 6[log6 (3y – 1)]

logb bx = x

log8 84 = 4

blogb x = x

6[log6 (3y – 1)] = 3y – 1Here are some special logarithm values:

1. loga 1 = 0 because a0 = 1

2. loga a = 1 because a1 = a

3. loga ax = x because ax = ax

How do you graph a logarithmic function?

Example: Graph f(x) = log3 x

This is the inverse of g(x) = 3x

We will need to create a table of values.(Keep in mind that logarithmic functions are inverses of exponential functions)

x g(x)

-2-1 0 1 2

1/91/3 1 3 9

x f(x)

-2-1 0 1 2

1/91/3 1 3 9

f(x) = log3 x

g(x) = 3x

A logarithmic function is the inverse of an exponential function.

For the function y = 2x, the inverse is x = 2y.

In order to solve this inverse equation for y, we write it in logarithmic form.

x = 2y is written as y = log2x and is read as “y = the logarithm of x to base 2”.

x -3 -2 -1 0 1 2 3 4

y 1

8

1

4

1

21 2 4 8 16

x

y -3 -2 -1 0 1 2 3 4

1

8

1

4

1

21 2 4 8 16

y = 2x

y = log2x

(x = 2y)

y = 2x

y = x

y = log2x

Graphing the Logarithmic Function

The y-intercept is 1.

There is no x-intercept.

The domain is {x | x R}.

The range is {y | y > 0}.

There is a horizontal asymptoteat y = 0.

There is no y-intercept.

The x-intercept is 1.

The domain is {x | x }.

The range is {y | y R}.

There is a vertical asymptoteat x = 0.

y = 2x y = log2x

The graph of y = 2x has been reflected in the line of y = x, to give the graph of y = log2x. This is because logarithmic functions are inverses of exponential functions

Comparing Exponential and Logarithmic Function Graphs

Logarithms

Consider 72 = 49.

2 is the exponent of the power, to which 7 is raised, to equal 49.

The logarithm of 49 to the base 7 is equal to 2 (log749 = 2).

72 = 49 log749 = 2

Exponential notation

Logarithmic form

In general: If bx = N, then logbN = x.

State in logarithmic form:

a) 63 = 216

b) 42 = 16

log6216 = 3

log416 = 2

State in exponential form:

a) log5125 = 3

b) log2128= 7

53 = 125

27 = 128

Evaluating Logarithms

1. log2128

log2128 = x 2x = 128 2x = 27

x = 7

2. log327

log327 = x 3x = 27 3x = 33

x = 3

Note:log2128 = log227

= 7 log327 = log333

= 3

3. log556 = 6 logaam = m

4. log816

log816 = x 8x = 16 23x = 24

3x = 4

5. log81

log81 = x 8x = 1 8x = 80

x = 0

loga1 = 0

x 4

3

6. log4(log338)

log48 = x 4x = 8 22x = 23

2x = 3

7. log 4 83

log 4 83 = x

4x 83

2 2x 23

3

2x = 1

8. 2 log2 8

2 log2 23

= 23

= 8

9. Given log165 = x, and log84 = y, express log220 in terms of x and y.log165 = x

16x = 5 24x = 5

log84 = y8y = 423y = 4

log220 = log2(4 x 5) = log2(23y x 24x) = log2(23y + 4x) = 3y + 4x

Evaluating Logarithms

x 3

2x

1

2

Base 10 logarithms are called common logs.

Using your calculator, evaluate to 3 decimal places:

a) log1025 b) log100.32 c) log102

1.398 -0.495 0.301

Evaluating Base 10 Logs

Your Task

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