8.3 – Logarithmic Functions and Inverses

What is a logarithm?

A logarithm is the power to which a number must be raised in order to get some other number

For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:

100 = 102 because log 10100 = 2

xy bx log then ,by If

UP, DOWN, UP

then ,by If x then ,37 If x

thenlog If b xy then364log If 4

3)8

1(log

2

1

Ex: Write the following equations in logarithmic form

Remember: If y = bx then logby = x

If 25 = 52

then

Log525=2

If 729 = 36

then

Log3729=6

If 1 = 100

then

Log101=0

If

then8

1

2

13

Let’s try some:Converting between the two forms.

Expo Form Log Formx7log3

364log4

210010 log

6443 225

15 log

32152

293 log

1000103

25

15 2

532

1log2

100102

73 x

932 31000log10

Let’s pause for a second . . .

If y = bx then logby = x

• x in the exponential expression bx is the logarithm in the equation logby=x

•The base b in bx is the same as the base b in the logarithm

NOTE: b does not =1 and must be greater than 0The logarithm of a negative number or zero is undefines.

Common Logs

A common log is a logarithm that uses base 10. You can write the common logarithm log10y as log y

Scientists use common logarithms to measure acidity, which increases as the concentration of hydrogen ions in a substance. The pH of a substance equals

ions.hydrogen of

ionconcentrat theis H where,Hlog

Evaluating Logarithms

Ex: Evaluate log816

Log816=x Write an equation in log form

16 = 8x Convert to exponential form

24 = (23)x Rewrite using the same base. In this case, base of 2

24 = 23x Power of exponents

4 = 3x Set the exponents equal to each other

x=4/3 Solve for x

Therefore, Log816=4/3

Evaluating Logarithms

Ex: Evaluate

Write an equation in log form Convert to exponential form

Rewrite using the same base. In this case, base of 2. Use negative

expos!

-5 = 6x Set the exponents equal to each other

x=-5/6 Solve for x

xx 6565

2222

1

Therefore,

321

64log

x6432

1

x32

1log64

6

5

32

1log64

Let’s try some

27log9 100log10

Evaluate the following:

Let’s try some

27log9 100log10

Evaluate the following:

Graphs of Logarithmic Functions

A logarithmic function is the inverse of an exponential function

In other words, y= 10x and y=log10x are inverses of each other.

Where is the line of reflection?

y = 10x

Y = log10x

Let’s try a more complicated one

Find the inverse of y=log5(x-1)+2

y=log5(x -1)+2 Start with the original function x=log5(y -1)+2 Switch the x and y x-2=log5(y -1) Subtract 2 from both sides y-1=5(x-2) Rewrite in y=abx form y=5(x-2)+1 Add 1 to both sides

The inverse of y=log5(x-1)+2 is y=5(x-2)+1

Let’s try some

Find the inverse of each function:

Y=log0.5x y=log5x2 y=log(x-2)

Hint: what is the base?

Solutions

Find the inverse of each function:

Y=log0.5x y=log5x2 y=log(x-2)

How does this help us?

We don’t have a way to graph a log, BUT we can graph an exponential function. If we can find the inverse of the log function and graph it, the graph of the log is simply the reflection of the exponential function.

Let’s start with an easy one: what is the inverse of y=log2x?

y=2x

Now, make a table of values:

Graph of y=log2x and y=2x

Make a table of values and graph y=2x

X y=2x

-3 0.125

-2 0.25

-1 0.5

0 1

1 2

2 4

3 8What is the domain and range of this function?

D: all real numbers. R: y>0

Graph of y=log2x and y=2x

Now, reverse the coordinates to graph y=log2x

x y=2x

0.125 -3

0.25 -2

0.5 -1

1 0

2 1

4 2

8 3

Prediction: what will it look like?

What is the domain and range of this function?

D: x>0. R: all real numbers

Graph of y=log2x and y=2x

Graphing both functions, we get this.

Where is the line of reflection?

y=x

Let’s try one:

Graph y=log3(x+3). Determine the domain and range.

Let’s try one:

Graph y=log3(x+3)

Let’s try one:

Graph y=log3(x+3)

Let’s try one:

Graph y=log3(x+3)

Determining the domain and range

Domain: x>0Range: all real

numbersy=log2x

y=log3(x+3)Domain: x>-3Range: all real

numbers

How can you predict the domain and range without graphing?

Write a sentence summarizing how to determine domain and range.

y=log3(x-2) y=log3(x)+1

Predict: what is the domain and range of y=log3(x-2) and y=log3(x)+1

Domain: x>2

Range: all real numbers

Domain: x>0

Range: all real numbers

Note: the +1 moves the parent graph up but does not affect the left or right translation, so the domain remains x>0