5.7 Properties of Logarithms
Definition of Logarithm
If x = am, then _____________________________.
Property
Product
Exponential
Logarithm
Quotient
Power Property
X
X
Power of a Power
X
Power of a Product
Change of Bases
Definition of Rational Exponents
Definition of Negative
Exponents
Example A: Rewrite 3 log(2) – log(5) as a single logarithm.
X
X
X
Example B: Write log 45 as a sum of two logs.
Example C: Rewrite the expression log(2x) using the power property.
Example D: Write, in terms of common logarithms, the exponent to which you’d raise 10 to get
35.07 289
164
Investigation: Properties of Logarithms
Step 1
Use your calculator to complete the table. Record the values to three decimal places.
Step 2
Look closely at the values for the logarithms in the table. Look for pairs of values that add up to third
value in the table. For example, add log 2 and log 3. Where can you find the sum in the table?
Record the equations that you find in the form log 2 + log 3 = log _____. (Hint: You
should find at least six equations.)
Step 3
Write a conjecture based on your results from Step 2.
Step 4
Use your conjecture to write log 90 as the sum of two logs. Do the
same for log 30 and log 72. Then use the table and your calculator to test your
conjecture.
Complete the following statement:
log a + log b = log _____
Step 5
Now find pairs of values in the table that subtract to equal another
value in the table. Record your results in the form log 9 – log 3 = log ____.
Log form
Log 2
Log 3
Log 5
Log 6
Log 8
Log 9
Log 10
Log 12
Log 15
Log 16
Log 25
Log 27
Decimal Form
Describe any patterns you see.
Complete the following statement: log a – log b = log ____.
Step 6
Now find values in the table that can be multiplied by a small integer to give another value in the table,
such as 3  log 2  log____ . Describe any patterns you see. You may want to think about different ways to express
numbers such as 25 or 27 using exponents.
Step 7
How do the properties you recorded in Steps 4-6 relate to the properties of exponents?
Step 8
Evaluate log26 on your calculator. Look at log6 and log2 on your table. Write a conjecture about how
you can use those values to evaluate log26. Verify your conjecture by evaluating log327 and other expressions that you
come up with.
Step 9
Complete the statement using your findings from step 8.
Change of base property:
Logba = __________________________
Name: ____________________________________
Homework 5.7 – Properties of Logarithms
1. Use the properties of logarithms to rewrite each expression as a single logarithm.
a.
log 21 – log 7
b. -4 log 2
c. -2 log 5 + 4 log 5
2. Write each expression as a sum or difference of logarithms ( or constants times logarithms). Simplify the result if
possible.
a.
a b
log
4
5 c
 3 abc 

 4 x 

3
c. log 
b. log 4 ( r  3 s  4 r 3 )
3. Determine whether each equation is true or false.
a.
log 8 
log 32
4
b. log59 – log52 = log54.5
c. log
1
1

5 log 5
f. log
1
 2 log 2 9
2 81
b.
log 4 
d.
2
log 8
3
e. log 3 5 
g. log 6  2 log 6
1
log 5
3
h. log315 – log35 = 1
4. Change the form of each expression below using definitions or properties of logarithms or exponents. Name each
definition or property you use.
a.
log r – log s
d. logbxm
r
s
g.  
b.
1
ab
c. qa+b
e. (cd)m
f. logbxy
h. cm/n
i.
m
log a x
log a y
5. Draw the graph of a function whose inverse is not a function. Carefully describe what must be true about the
graph of a function if its inverse is not a function.
6. Find an equation to fit each set of data.
X
1
4
6
7
Y
8
17
23
26
X
0
3
4
6
Y
2
54
162
1458
Practice 5.7
1. Use the properties of logarithms to rewrite teach expression as a single logarithm.
a.
log5 + log11
b. 3log2
d. -2log6
c. log28 – log7
e. log7 + 2log3
2. Rewrite each expression as a sum or difference of logarithms by using the properties of logarithms.
a. Write log22 as a sum of two logs
b. Write log 13 as a difference of two logs.
c. Write log 29 as a sum of two logs.
d. Write log 7 as a difference of two logs.
3. Use the power property of logarithms to rewrite each expression.
a.
log5x
b. log x2
c. log 3
d. 2log 7x
4. Determine whether each equation is true or false. If false, rewrite one side of the equation to make it true. Check
your answer on your calculator.
a.
Log 3 + log 7 = log 21
b. log 5 + log 3 = log 8
c. log 16 = 4log2
d. log 5 – log 2 = log 2.5
e. log 9 – log 3 = log 6
f. log 7  log
g. log 35 = 5 log 7
h. log
j. log 64 = 1.5log16
l.
1
  log 4
4
log 7
 log 7 3
log 3
i.
log 3
3
 log
log 4
4
7
2
5. Change the form of each expression below using properties of logarithms or exponents. Name each property or
definition you use.
a.
gh+k
f. log bg
b. log s + log t
c.
fw
fv
d. log
km
h.
log s t
log s u
i. wtws
g.
n
h
k
e. (js)t
j. p-h
6. The half-life of carbon-14, which is used in dating archaeological finds, is 5730 yr.
a.
Assume that 100% of the carbon-14 is present at time 0 yr, or x = 0. Write the equation that expresses the
percentage of carbon-14 remaining as a function of time.
b. Suppose some bone fragments have 25% of their carbon-14 remaining. What is the approximate age of the
bones?
c. In the movie “Raiders of the Lost Ark” (1981), a piece of the Ark of the Covenant found by Indiana Jones
contained 62.25% of its carbon-14. What year would this indicate that the ark was constructed in?
d. Coal is formed from trees that lived about 100 million years ago. Could carbon-14 dating be used to determine
the age of lump of coal? Explain your answer.
7. Use the properties of logarithms and exponents to solve these equations.
a.
5.1x = 247
d. 23 + 45(1.024x) = 147
b. 17 + 1.25x = 30
c. 27(0.93)x = 12
Name: _______________________________________
Properties of Logarithms – Exit Ticket
Check off True/False where applicable. (Do not use a calculator)
𝑙𝑜𝑔100 is equivalent to…
True/False
1.
𝑙𝑜𝑔90 + 𝑙𝑜𝑔10
____/____
2.
𝑙𝑜𝑔10 + 𝑙𝑜𝑔10
____/____
3.
2 ∙ 𝑙𝑜𝑔10
____/____
4.
𝑙𝑜𝑔100 − 𝑙𝑜𝑔1
____/____
5.
𝑙𝑜𝑔2 + 𝑙𝑜𝑔10 + 𝑙𝑜𝑔5
____/____
6.
1
∙ 𝑙𝑜𝑔10,000
2
10 ∙ 𝑙𝑜𝑔10
____/____
log 3 100
log 3 10
____/____
7.
8.
____/____
Name: _______________________________________
Check off True/False where applicable. (Do not use a calculator)
𝑙𝑜𝑔100 is equivalent to…
True/False
1.
𝑙𝑜𝑔90 + 𝑙𝑜𝑔10
____/____
2.
𝑙𝑜𝑔10 + 𝑙𝑜𝑔10
____/____
3.
2 ∙ 𝑙𝑜𝑔10
____/____
4.
𝑙𝑜𝑔100 − 𝑙𝑜𝑔1
____/____
5.
𝑙𝑜𝑔2 + 𝑙𝑜𝑔10 + 𝑙𝑜𝑔5
____/____
6.
1
∙ 𝑙𝑜𝑔10,000
2
10 ∙ 𝑙𝑜𝑔10
____/____
log 3 100
log 3 10
____/____
7.
8.
____/____
Properties of Logarithms – Exit Ticket