4.4 Evaluate Logarithms and Graph
Logarithmic Functions
Part 2
Definition
• Logarithms are the "opposite"
of exponentials,
• Logs "undo" exponentials.
• Logs are the inverses of exponentials.
Writing Logarithms
_____________________________________________
log b a  c
-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ____________________________________________
You read it: Log base “b” of “a” equals “c”
‘log’
b
a
c
is the operation
is the base
is the object of the log
is what you get when you evaluate the log
Exponential Form
x
b =y
Logarithmic Form
log x y
b =
Evaluating logarithms now
you try some!
•
•
•
•
4  16
Log 4 16 = 2
x
0
Log 5 1 =
5 1
x
Log 16 4 = ½ (because
16  16
4 1/2 = 4)
x
undefined
3  1
Log 3 (-1) =
x
(Think of the graph of y = 3x)
You should learn the
following general forms!!!
• Log a 1 = 0 because a0 = 1
• Log a a = 1 because
1
a
=a
• Log a ax = x because ax = ax
Common logarithms
•log x = log 10 x
• Understood base 10 if
nothing is there.
Common Logs and
Natural Logs with a
calculator
log10 button
lne button
Finding Inverses
• Find the inverse of:
• y = log3x
• By definition of logarithm, the inverse is
y=3x
• OR write it in exponential form and switch
the x & y!
y
3
=x
x
3
=y
Example 1:
• Write 53 = 125 in logarithmic form.
• Write log381 = 4 in exponential form.
Try This:
Complete the table.
#1
Exponential
Form
Logarithmic
Form
#2
25 = 32
#3
#4
3-2 = 1/9
log101000 = 3
Log164 = 1/2
Lets look at their graphs
y  10
x
y  log10 x
log10 y  x
y=x
To Evaluate Logs without a
Calculator
• Change the log to an exponential.
1.
log2 32 = x
2.
log4 2 = x
Solve for x.
Change the log to an exponential.
1.
log2 64 = x
2.
logx 343 = 3
Evaluate without a calculator:
Change the log to an exponential.
1.
log 2 8 = x
2. log 2 1 = x
3. Find the value of k :
k = log
4. Find the value of k :
½ = log k 9
5. Find the value of k :
3 = log 7 k
9
3
Common Logarithms
10 are called
• Logarithms with base ______
common logarithms.
• Sometimes the base is assumed and not
written.
• Thus, if you see a log written without a base,
10
you assume the base is _______.
• The log button the calculator uses base
10
_____.
Use your calculator to
evaluate:
1. log 51
1.71
2. log 4
0.6
3. log 0.215
– 0.67
Which means
101.71  51
Do You Know What X is?
Change the exponential to a log.
Then use calculator.
4. Solve for x:
10x = 728
5. Solve for x:
1
10 
1085
x
Remember
e?
e  2.718
Natural Logarithm
• A natural logarithm is a logarithm with base e,
denoted by ln.
• A natural logarithm is the inverse of an exponential
function with base e.
log e x  ln x
Exponential Form
e  7.389
2
Logarithmic Form
ln 7.389  2
Lets look at their graphs
ye
y  ln  x 
x
ln y  x
y=x
Write as exponent or log.
y  ln x
1. e  4
x
2. ln 56.3  4.03
Evaluate f(x)=ln x to the nearest
thousandth for each value of x below:
3. x  2
0.693
1
4. x 
2
– 0.693
5. x  1
?
(see graph)
13. Find the inverse of y = ln(x+1)
y = ex - 1
14. Find the inverse of y = 5x .
y = log5x
Homework
Book
Pg. 147 16 - 24 all
Pg. 148 13 – 21 all