5.3 Intro to Logarithms
2/27/2013
Definition of a Logarithmic Function
For y > 0 and b > 0, b ≠ 1,
logb y = x
if and only if bx = y
Note: Logarithmic functions are the inverse of
exponential functions
Example: log2 8 = 3 since 23 = 8
Read as: “log base 2 of 8”
Location of Base and Exponent in
Exponential and Logarithmic Forms
Exponent
Exponent
Logarithmic form: x = logb y
Base
Exponential Form: bx = y
Base
Basic Logarithmic Properties
Involving One
logb y = x
if and only if bx = y
1
• Logb b = __
because 1 is the exponent to which b must be raised to obtain b.
(b1 = b).
0
• Logb 1 = __
because 0 is the exponent to which b must be raised to obtain 1.
(b0 = 1).
Popular Bases have special names
Base 10
log 10 x = log x
Base “e”
log e x = ln x
is called a common logarithm
is called the natural logarithm
or “natural log”
e and Natural Logarithm
e is the natural base and is also called “Euler’s
number”
: an irrational number (like ) and is
approximately equal to 2.718281828...
Real Life Use: Compounding Interest problem
Remember the formula 𝐴 𝑡 = 𝑃𝑜 1 +
+
𝑟 𝑛𝑡
𝑛
1+
1 𝑛
=
𝑛
𝑒 as n approaches
The Natural logarithm of a number x (written as
“ln (x)”) is the power to which e would have to
be raised to equal x.
For example, ln(7.389...) is 2, because e2=7.389
Note: 𝑒 𝑥 and ln(x) are inverse functions.
Inverse properties
Since 𝑏 𝑥 𝑎𝑛𝑑 log 𝑏 𝑥 are inverse functions.
Proof: log 2 8 = 3
Then 23 = 8
2log2 8 = 8
𝑏 log𝑏 𝑥 = 𝑥
𝑥
and log 𝑏 𝑏 = 𝑥
Proof: 32 = 9
log 3 9 = 2
log 3 32 = 2
Since 𝑒 𝑥 and ln(x) are inverse functions.
𝑒 ln(𝑥) = 𝑥
and
𝑥
ln (𝑒 ) = 𝑥
Example 1
Rewrite in Exponential Form
logb y = x
LOGARITHMIC FORM
a. log2 16 = 4
is bx = y
EXPONENTIAL FORM
24 = 16
b. log7 1
=
0
70
=
1
c. log5 5
=
1
51
=
5
10 – 2
=
d. log 0.01 =
e. log1/4 4
–
2
= –1
1
–1
=
4
0.01
4
Example 1
Rewrite in Exponential Form
EXPONENTIAL FORM
LOGARITHMIC FORM
log e x = ln x
f. ln 𝑒 2 = 2
𝑙𝑛 𝑒 2 = log 𝑒 𝑒 2 = 2
g. ln 𝑥 = 2
ln 𝑥 = log 𝑒 𝑥 = 2
𝑒2= 𝑒2
𝑒2= 𝑥
Example 2
Rewrite in Logarithmic Form Form
logb y = x
EXPONENTIAL FORM
b. 10−2 = .01
log .01 = −2
log 𝟔 1 = 0
c. 60 = 1
d.
LOGARITHMIC FORM
log 𝟕 49 = 2
a. 72 = 49
10−4 =
is bx = y
1
10,000
1
log
= −4
10000
Example 3
Evaluate Logarithmic Expressions
Evaluate the expression.
a. log4 64
logb y = x
is bx = y
4? = 64
What power of 4 gives 64?
43 = 64
Guess, check, and revise.
log4 64 = 3
b. log4 2
4? = 2
What power of 4 gives 2?
41/2 = 2
Guess, check, and revise.
1
log4 2 =
2
Example 3
Evaluate Logarithmic Expressions
c. log1/3 9
1
3
1
3
?
1
=9
What power of
=9
Guess, check, and revise.
3
gives 9?
–2
log1/3 9 = – 2
d. ln 𝑒 4
Since 𝑙𝑛 (𝑒 𝑥 ) = 𝑥
𝑙𝑛 𝑒 4 = 4
Example 4
a.
Simplifying Exponential Functions
7log7 5
Since 𝑏 log𝑏 𝑥 = 𝑥
7log7 5 = 5
b.
2log2
3
Since 𝑏 log𝑏 𝑥 = 𝑥
2log2
3
= 3
Example 4
c.
Simplifying Exponential Functions
𝑒 ln 6
Since 𝑒 ln(𝑥) = 𝑥
𝑒 ln 6 = 6
d.
𝑒
ln 𝑥 3
Since 𝑒 ln(𝑥) = 𝑥
3
ln
𝑥
𝑒
=
𝑥3
Homework
WS 5.3 odd problems only