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PreCalculus
Section 3-3 Properties of Logarithms
Name: ______________________
Period _____
Properties of Logarithms: Key Concept on page 181
Let b, x and y be positive numbers and b ≠1.
Condensed form
Expanded form
Product
Property
If there is multiplication inside one log,
you can split it up as addition of two logs.
logb xy  logb x  logb y
Quotient
Property
Power
Property
If there is division inside one log,
you can split it up as subtraction of two logs.
logb
The logarithm of a power is equal to the
product of the exponent and the logarithm.
x
 logb x  logb y
y
log b x p  p log b x
Ex. 1: Express each logarithm in terms of log 2 and log 3.
32
a. log 96
b. log
9
c. log 72
Ex. 2: Evaluate each logarithm without using a calculator.
a. log2 3 32
b. 3ln e4  2 ln e2
Ex. 3: Expand each expression.
a. ln 4m3 n5
b. log
Ex. 4: Condense each expression.
1
a. log 4 x  3log 4  x  2 
2
PreCalculus Ch 3B Notes_Page 1
2x  3
34 x
b. 5 ln (x+1) + 6 ln x
c. ln
3y  2
43 y
1
1
c. 2 log 3 x  log 3 y  log 3 (3 y  1)
2
5
Change of Base Formula: See key concept on page 183.
This formula allows you to evaluate any logarithm using a calculator.
--- For any positive real numbers, a, b, and x, a ≠ 1 and b ≠ 1,
log a x
log x
ln x
log b x 
In particular, logb x 
and logb x 
log a b
log b
ln b
Ex. 5: Evaluate each logarithm using change of base formula.
a. log64
b. log 1 8
3
Ex. 6: Diversity in a certain ecological environment containing two different species is modeled by the
N N
N 
N
function D    1 log 2 1  2 log 2 2  , where N1 and N2 are the numbers of each type of species
S
S
S 
 S
found in the sample and S = (N1 + N2). Find the measure of diversity for the environments that find the
following number of species in the samples.
a.
25 and 50
b.
10 and 60
PreCalculus Ch 3B Notes_Page 2
PreCalculus
Section 3-4 Exponential and Logarithmic Equations
Name: ___________________
How to Solve Exponential Equations:
 1st method: Write both sides using the same base if possible.
(apply One-to-One Property of Exponential Functions)
For b > 0 and b ≠ 1, bx = by if and only if x = y.
Example: If 3x  35 , then x = 5. If x = 5, then 3x  35 .
 2nd method: Take a logarithm of each side.
How to Solve Logarithmic Equations:
 1st method: If the equation is “single log  single log, ”
then apply One-to-One Property of Logarithmic Functions.
For b > 0 and b ≠ 1, logb x  logb y if and only if x = y.
Example: If log 2 x  log 2 7 , then x = 7.
 2nd method: If the equation is “single log  a number ” rewrite as an exponential equation.
(or exponentiate each side of an equation)
*Extraneous Solutions: Because the domain of a logarithmic function y= log b x is x  0 ,
you need to check for extraneous solutions of any logarithmic equation.
Ex. 1: Solve each equation. (Use bx = by if x = y)
n
a. 4
x2
 16
x 3
2
 1   1 3
b.     
 3   81 
Ex. 2: Solve each equation.
a. 2 ln x = 18
d. 4  3log(5 x)  16
PreCalculus Ch 3B Notes_Page 3
c. 8
x4
 32
b. 7 – 3 log 10x = 13
e. log 3 ( x 2  1)  4
3x
d.
2
 
3
x5
3x
94
 
 4
c. log 5 x 4  20
Ex. 3: Solve each equation. (Use logbx = logby, if x = y)
a. log2 5  log2 10  log 2  x  4 
b. log5  x 2  x   log5 20
c. log12  x  3  log12 x  log12 4
Ex. 4: Solve each equation, round to the nearest hundredth.
a. 3x = 7
b. e2x+1 = 8
Ex. 5: Solve each equation, round to the nearest hundredth.
a. 36x-3 = 24-4x
b. 62x+4 = 5–x+1
PreCalculus Ch 3B Notes_Page 4
Ex. 6: Solve each equation, round to the nearest hundredth.
a. e2 x  e x  2  0
b. 4e4 x  8e2 x  5
Ex. 7: Solve each equation. Check for extraneous solutions.
a. log x + log(x -3) = log 28
PreCalculus Ch 3B Notes_Page 5
b. ln (7x+3) – ln (x+1) = ln (2x)
Ex. 8: Solve each equation. Check for extraneous solutions.
a. log(3x - 4) =1 + log (2x + 3)
b. log (x - 12) = 2 + log (x - 2)
Ex. 9: The table below shows the number of cell phones a new store sold in March and August, 2014.
Cell Phones
Month
Number Sold
March
88
August
177
PreCalculus Ch 3B Notes_Page 6
(a) If the number of phones sold per month is increasing at an exponential
rate, identify the continuous rate of growth. Then write the exponential
equation to model this situation.
(b) Use your model to predict the number of months it will take for the store
to sell 500 phones in one month.