Upload
britney-osborne
View
216
Download
1
Embed Size (px)
Citation preview
Logarithmic Functions
The inverse of the equation
y = bx is x = by
Since there is no algebraic method for solving x = by for y in terms of x, the Logarithmic Function is used to allow y to be expressed in terms of x.
That’s right! Interchange x and y.
Sounds pretty easy so far. Let’s move on.
Let’s Take a Closer Look at Some Logs
A logarithm is really an exponent written in a different form.
The equation y = bx is an exponential function
Let’s break this down.
b is the basex is the exponenty is the value of bx
Now let’s bring in the logs.
Written in logarithmic form, the equation y = bx
would bex = log b aWe read this
asx is the logarithm of a with base b
Breaking Down Logs
Let’s look at a log piece by piece.
The equation x = log b
ais a logarithmic function
Let’s break this down.
b is the basex is the exponenta is the value of bx
Hey! I’ve seen this before.
It’s Sam Ting
as breaking down exponential functions.
That was easy
Comparing Logarithmic form and exponential
form
Exponential Form Logarithmic Form
y = bx
x = log b a
32 = 25 5 = log 2 32
512 = 83 3 = log 8 512
4 = log 3 81
3 = log 5 125
81 = 34
125 = 53
AsiDe
Facil
Logarithms with Variables
3 = log 4 a
In each equation, find the value of the variable
since 43 = 64, a = 64
x = log 6
36
since 62 = 36,
x = 2
3 = log b 125
since 53 = 125,b = 5
Hey, I can just use my calculator for this.
This looks a little harder. Maybe I should use a real calculator for this one.
That was easy
43 = a 6x = 36 b3 = 125
More Logarithms with Variables
In each equation, find the value of the variable
5 = log 8 a
since 85 = 32,768, a = 32,768
x = log 7 2,401
since 74 = 2,401,x = 4
3 = log b
6,859
since 193 = 6,859,b = 19
Hey, those are some pretty big numbers. I hope my calculator knows how to do this.
That was easy
85 = a 7x = 2,401 b3 = 6,859
Common Logs
Any logarithm with base 10 is a Common Log
When writing a common logarithm, the base is usually omitted.
So, 5 = log 10 100,000 and 5 = log 100,000 are Sam Ting.
Let’s compare Logarithmic Form and Exponential Form of some Common Logs.
Exponential FormLogarithmic Form
3 = log 1,000 1,000 = 103
1,000,000 = 1066 = log 1,000,000
10,000 = 104
4 = log 10,000
That was easy
Common Logs with Variables
In each equation, find the value of the variablex = log 100
10x = 100
since 102 =
100, then x = 2count the zeros
7 = log a
107 = a
since 107 = 10,000,000, then a = 10,000,000
write the proper number of zeros
Hey, I don’t even need a calculator for this!
That was easy
More Common Logs with Variables
Find the value of the variable to the nearest one hundredthx = log 1,345
10x = 1,345
Hey, there’s no zeros to count.
2.865 = log a
102.865 = a
That was easy
We could use the LOG key on our calculator.
LOG (1,345) = 3.13
What’s the proper number of zeros?We could use the 10x key on our calculator.102.865 = 732.82
Change of BaseHow can I get my calculator to evaluate logs in bases other than base 10? That’s easy, just use
the Change of Base Formula
logbx a
x = log 8 512 = 3
x = log 12 248,832 = 5
It’s time to push the easy button once again!
loglog
ab
log512
log8
248log ,832log12
More Change of Base
Let’s throw some decimals into the mix.
x = log 4 32
x = log 4.5 91.125 = 3
= 2.5
This stuff is too easy. Soon I’ll have to buy new batteries for my easy button.
x = log 8.125 1,986.597 = 3.625
That was easy
91log .125log4.5
log32
log4
1,98log 6.597log8.125