Download Math 152B - Cass Gustafson – 9.5 Exponential and Logarithmic

Math 152B - Cass
Gustafson – 9.5
Exponential and Logarithmic Functions
Properties of Logarithms
I.
Basic Logarithmic Properties Based on the Definition of a Logarithm
A. If b > 0 and b ≠ 1, then
1) logb b = 1 because …
2) logb b x = x because …
3) logb 1 = 0 because …
B. If b = 10, then
1) log 1 = 0 because …
2) log 10 = 1 because …
C. If b = e, then
1) ln 1 = 0 because …
3) ln e = 1 because …
4) blogb x = x because …
x
3) log 10 = x because …
4) 10
log x
= x because …
x
2) ln e = x because …
4) e
ln x
= x because …
Ex. 1: Evaluate the following. Hint: set an expression equal to x if you’re not sure what rule applies.
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a) log9 9
b) log 10
c) log8 1
d) ln e
e) 3log3 17
f)
g) log7 78
h) 10
ln 1
log 7
II. Properties of Logarithms Based on the Properties of Exponents
Since a logarithm is an exponent, the properties of exponents are also properties of logarithms.
A. The Product Rule: The logarithm of a product is the sum of the logarithms.
logb ( MN ) = logb M + logb N
for positive real numbers M, N, and b, b ≠ 1.
Ex. 2: Expand the following expressions using the Product Rule.
a) log6 ( 7 ⋅11)
b) log (100x )
B. The Quotient Rule: The logarithm of a quotient is the difference of the logarithms.
M 
logb   = logb M − logb N
N
for positive real numbers M, N, and b, b ≠ 1.
Ex. 3: Expand the following expressions using the Quotient Rule.
 23 
a) log8 

 x 
 e5 
b) ln  
 11 
C. The Power Rule: The logarithm of a number with an exponent is the product of the exponent
and the logarithm of that number.
logb Mp = p ⋅ logb M
for positive real numbers M and b,
b ≠ 1, and any real number p
Ex. 4: Expand the following expressions using the Power Rule.
a) log6 89
Gustafson – 9.5
b) ln 3 x
2
III. Expanding Logarithmic Expressions: When we use the above rules to write a single logarithm as a
sum, difference, or product of other numbers and/or logarithms, we say that we are expanding a
logarithmic expression.
Ex. 5: Use the logarithmic properties to expand each expression and write it in terms of the
logarithms of x, y, and z.
 x 
a) logb  
 yz 
 x3 y2
b) logb  4 4
 z





IV. Condensing Logarithmic Expressions: When we use the rules of logarithms to write the sum or
difference of two or more logarithmic expressions as a single logarithmic expression, we say that we
are condensing a logarithmic expression.
Ex. 6: Write as a logarithm of a single quantity.
a) −2 logb x − 3 logb y + logb z
b) 2 logb x +
1
log y − 2 logb ( x − y )
2 b
Ex. 7: Given that log 5 ≈ 0.6990 and log 6 ≈ 0.7782, find approximations for the following without
using a calculator.
a) log 30
b) log 25
c) log 180
 1
d) log  
2
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V. The Change-of-Base Property: The logarithm of M with base b is equal to the logarithm of M with
any new base divided by the logarithm of b with that new base.
logb M =
loga M
for positive real numbers M, a and b, a ≠ 1, b ≠ 1.
loga b
logb M =
logM
lnM
=
for positive real numbers M and b, b ≠ 1.
logb
lnb
Ex. 8: Find each logarithm to four decimal places.
a) log7 2506
b) logπ e
VI. Properties of Equality for Exponentials and Logarithms
These properties will be used when solving exponential and logarithmic equation.
M
A. b = b
B. M = N
N
M = N (for b > 0 and b ≠ 1)
logb M = logb N (for all positive M and N)
VII. Applications
A. pH of a Solution: The pH of a solution is a measure of its acidity, that is, a measure of the
concentration of hydrogen ions in the solution. The more acidic a solution, the greater the
concentration of hydrogen ions. The concentration is indicated by the pH scale, which is defined
by the following equation…
pH = –log [H+] , where [H+] is the hydrogen ion concentration in gram-ions per liter.
Ex. 9: Find the hydrogen ion concentration of a saturated solution of calcium hydroxide whose
pH is 13.2.
B. In physiology, the relationship between the loudness and the intensity of sound is based on the
Weber-Fechner law, which is defined by the following equation…
L = k ln I , where L is the apparent loudness of a sound, I is the actual intensity, and k is a
constant.
Ex. 10: If the intensity of a sound is tripled, find the apparent change in loudness.
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