Logarithm Notes
Definition
As a function
(a is the base, y is the exponent)
As an equation
(b is the base, x is the exponent)
Logarithmic
Form
y = loga x
example: y = log5 x
logb a = m
example: log5 25 = x
Exponentia
l form
x = ay
example: x = 5y
bm = a
example: 5x = 25
Facts about Logarithms
1. Invented in early 1600’s. Has numerous applications in finance, math and science.
Prior to calculators the slide rule, based on logarithms, allowed for quick arithmetic
computations.
2. Is an exponent
3. Used to solve exponential equations
4. Log functions are inverses of exponential functions
a.
y = logax is the inverse of y=ax
b. example: y = log5 x is the inverse of y = 5x.
i. To illustrate this exchange x & y, we get x = log5 y, so y = 5x
Properties
Example
1. logb b = 1 & logb 1 = 0 & logb 0 =
log7 7 = 1 and log7 1 = 0 and log7 0 = O
log7 15 = log7 5 + log7 3
2. (product rule)
logb xy = logb x + logb y
3. (quotient rule)
x
logb y = logb x - logb y
4. (power rule)
logb xr = r logb x
log7
log7
5
= log7 5 - log7 3
3
3
1
1
5 = log7 5 3 = log7 5
3
Special case log7 75 = 5
Special case logb bx = x
5. Properties used to solve log
equations:
a. if bx = by, then x = y
b. if logb x = logb y, then x = y
a. if 3x + 3 = 34, then x + 3 = 4
x=1
b. if log7 x + 3 = log7 4, then x + 3 = 4
Notes:
i. x > 0 and y >0 must be true
ii. converse is also true
Natural Logarithms
Definition
Value of e
ln x = loge x
Function
Inverse
y = ln x
y = ex to illustrate switch x & y so x = ln y = loge y and y = ex 
ln (ex) = x to illustrate loge ex = x loge e = x
All of the above properties apply
Property
Other Properties
e1  2.72 also defined as
1111
  



0
!1
! 2
! 3
!
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x=1
PROOF OF PROPERTIES
Property
Proof
1. logb b = 1 and logb 1 = 0
2. (product rule)
logb xy = logb x + logb y
b = b  and b = 1 
a. Let logb x = m and logb y = n
b. x = bm and y = b n
c. xy = bm * bn
d. xy = b m + n
e. logb xy = m + n
f logb xy = logb x + logb y 
Definition of logarithms
a. Setup
b. Rewrite in exponent form
c. Multiply together
d. Product rule for exponents
e. Rewrite in log form
f. Substitution
3. (quotient rule)
x
logb y = logb x - logb y
a. Let logb x = m and logb y = n
b. x = bm and y = b n
x
bm
c. y = n
b
x
m-n
d. y = b
x
e. logb y = m - n
x

f. logb y logb x - logb y 
a. Given: compact form
b. Rewrite in exponent form
4. (power rule)
logb xn = n logb x
1
a.
b.
c.
d.
Reason for Step
0
Let m = logb x so x = bm
xn = bmn
logb x n = mn
logb xn = n logb x
c. Divide
d. Quotient rule for exponents
e. Rewrite in log form
f. Substitution
a.
b.
c.
d.
Setup
Raise both sides to the nth power
Rewrite as log
Substitute
5. Properties used to solve
log equations:
a. if bx = by, then x = y
a. This follows directly from the properties
for exponents.
b. if logb x = logb y, then x = y
b. i. logb x - logb y = 0
x
0
y
ii. logb
x
iii. y = b0
x

iv. y 1 so x = y
b. i. Subtract from both sides
ii. Quotient rule
iii. Rewrite in exponent form
iv. b0 = 1
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