MHF 4U – Unit 7: Exponential and Logarithmic Functions
Working with Logarithms
Date: _____________
x = 10 y is the equation of the inverse of y = 10 x in exponential form.
y = log10 x is the equation of the inverse of y = 10 x in logarithmic form.
A base of 10 is usually not
written (like an exponent of 1 on
x is not usually written).
We usually write y = log x
This equation can be interpreted as:
y = " what do I have to raise ten to the power of to get x"
or “what exponent do I put on ten to get x”
For example, if x = 100, then
y = log100
because 10 2 = 100
y=2
According to this then, if y = b x then x = b y is the inverse in exponential form and y = log b x is the
inverse in logarithmic form.
Evaluating simple logarithmic expressions
If y = log b x
y = " what do I have to raise b to the power of to get x" or “what exponent do I put on b to get x”
then log 2 2 is asking “what exponent do you put on 2 to get 2?” ∴ log 2 2 = 1
log 2 4 is asking “what exponent do you put on 2 to get 4?” ∴ log 2 4 = 2
and log 2 16 = 4
Look at the following example:
Evaluate
1
9
∴ log 3 19 = −2 _
c) log 3
_ 53 = 125 ∴ log 5 125 = 3 _ _
_ 3 −2 =
_ 2 3 = 8 ∴ log 2 8 = 3 _ _
d) log 25 5
1
9
e)
1
2
_ ∵ 25 = 5, 25 = 5 ∴ log 25 5 =
1
2
_ 4
_
− 12
=
1
2
∴ log 4 12 =
−1
2
_
By flopping back and forth between exponential and logarithmic form we can solve simple exponential
and logarithmic equations.
For Example:
x = log100
3 = log x
4 = log x 81
x = log10 100
3 = log10 x
⇔ x 4 = 81
⇔ 10 x = 100
10 x = 10 2
∴x = 2
x 4 = 34
∴x = 3
⇔ 10 3 = x
x = 1000
Page 1 of 6.
Recall: Laws of exponents
Example: express as a single power and/or
evaluate
Type
Law
Products
(a )(a )=
am+n
(2 )(2 )= 28
Quotients
a m ÷ a n = am−n
5 3 ÷ 5 7 = 5 −4
Powers
(a )
Zero exponents
a0 = 1
m
n
m n
= a m⋅n
3
5
4 5
(6 ) =
6 20
1010 = 1
n
−n
1
1
=   → n
a
a
Negative exponents
a
Power of a product
(ab)
Power of a quotient
 a m a m
  = m
b
b
m
= a m ⋅ bm
1
32
3−2 =
6
(2 ⋅ 5)
 2 6 26
  = 6
5
5
3
m
n
Rational exponents
n
a = a
or
( a)
n
25 2 =
m
= 2 6 ⋅ 56
a
m
( 25 )
2
3
= 53
= 125
Logarithms follow similar rules. Let’s investigate them!
Investigation A
1) Explore logarithms to the base 10. Note that log x = log10 x .
2) Explore logarithms to the base 10. Note that log x = log10 x .
a) Use a calculator to evaluate the following
logarithms (all to base 10):
c) Use a calculator to evaluate the following
common logarithms:
Log 4 = 0.60206
Log 1 = 0
Log 40 = 1.60206
Log 10 = 1
Log 400 = 2.60206
Log 100 = 2
Log 4000 = 3.60206
Log 1000 = 3
b) What would you expect the answer to be for
log 40 000? _4.60206_________
d) You can express log 40 as a logarithm of a product, log(10 ⋅ 4 ). How would you express log 40 in
terms of log 10 and log 4 ?
log 40 = log (10 ⋅ 4 )
= log 10 + log 4
MHF 4U Unit 7 - Laws of Exponents and Logarithms
Page 2 of 6.
e) Rewrite each of the following as a product of two logs from a) and c):
log 400 = log (100 ⋅ 4 ) _ _
log 4000 = log (100 ⋅ 40 ) _ _ _
= log 100 + log 4 _ _ _
= log 100 + log 40 _ _
2) Use your knowledge from the previous lesson to evaluate logarithms with bases other than 10, and
complete the following table.
Evaluate:
Evaluate:
log 2 4 = 2
log 2 8 = 3
log 3 9 = 2
log 3 27 = 3
Use your results from #1 and the above values
to determine the value of
log 2 (4 ⋅ 8) = log 2 4 + log 2 8
= 2+3
=5
Use your results from #1 and the above values
to determine the value of
log 3 (27 ⋅ 9 ) = log 3 27 + log 3 9
= 3+ 2
=5
Verify by multiplying first:
log 2 (4 ⋅ 8) = log 2 32
=5
Verify by multiplying first:
log 3 (27 ⋅ 9 ) = log 3 243
=5
3) How could we rewrite the general case, log a ( xy ) ?
log a ( xy ) = log a x + log a y
4) If log 24 = a and log 4 = b, determine an
expression for log 96 in terms of a and b?
log 96 = log( 24 ⋅ 4 )
log 96 = log( 24 ) + log( 4 )
log 96 = a + b
Investigation B
1) Explore logarithms to the base 10. Remember that log x = log10 x .
a) Use a calculator to evaluate the following
logarithms (all to base 10):
Log 40 000 = 4.60206
Log 4000 = 3.60206
Log 400 = 2.60206
Log 40 = 1.60206
c) Use a calculator to evaluate the following
common logarithms:
Log 1000 = 3
Log 100 = 2
Log 10 = 1
Log 1 = 0
b) What would you expect the answer to be for
log 4? 0.60206_______
d) You can express log 4 as a logarithm of a quotient, log(40 ÷ 10) . How would you express log 4 in
terms of log 10 and log 40 ?
log 4 = log (40 ÷ 10 )
= log 40 − log 10
MHF 4U Unit 7 - Laws of Exponents and Logarithms
Page 3 of 6.
e) Rewrite each of the following in terms of a quotient of two logs from a) and c).
log 4000 = log (40000 ÷ 10)
log 400 = log (4000 ÷ 10)
= log 40000 − log 10
= log 400 − log 10
2) Use your knowledge from the previous lesson to evaluate logarithms with bases other than 10, and
complete the following.
Evaluate:
Evaluate:
log 2 64 = 6
log 2 16 = 4
log 4 16 = 2
log 4 64 = 3
Use your results from #1 and the above values
Use your results from #1 and the above values
to determine the value of
to determine the value of
 16 
log 2 (64 ÷16) = log 2 64 − log 2 16
log 4   = log 4 16 − log 4 64
 64 
=6−4
= 2−3
=2
= −1
Verify by dividing first:
Verify by dividing first:
 16 
log 2 (64 ÷ 16) = log 2 4
log 4   = log 4 14
=2
 64 
= −1

3) How could we rewrite the general case log a 

x
?
y
4) If log 96 = a and log 4 = b, determine an
expression for log 24 in terms of a and b?
log 24 = log( 96 ÷ 4 )
log 24 = log( 96 ) − log( 4 )
log 24 = a − b
x
log a   = log a x − log a y
 y
Investigation C
1) Evaluate each power then use your knowledge from the previous lesson to evaluate logarithms
with bases other than 10. Complete the following table
Evaluate:
Evaluate:
Evaluate:
log 3 3 = 1
log 2 4 = 2
log 4 16 = 2
1
log 3 32 = 2
log 2 4 3 = 6
log 4 16
How are the two values related?
How are the two values related?
How are the two values related?
log 3 3 = 2(log 3 3)
log 2 43 = 3(log 2 4 )
Evaluate:
log 5 125 = 3
Evaluate:
log 4 64 = 3
2
log 5 125
1
3
= 1
How are the two values
related?
1
1
log 5 125 3 = log 5 125
3
log 4 64
1
6
= 1/2
How are the two values
related?
1
1
log 4 64 6 = log 4 64
6
MHF 4U Unit 7 - Laws of Exponents and Logarithms
log 4 16
1
2
2
=1
=
1
(log 4 16)
2
Evaluate:
log 3 81 = 3
log 3 81
1
2
= 3/2
How are the two values
related?
1
1
log 3 81 2 = log 3 81
2
Page 4 of 6.
2) How could we rewrite the general case, log a m n ?
log a m n = n ⋅ log a m
Investigation D
1) a) Evaluate log 7 49 = 2
since 72 = 49
b) Use your calculator to evaluate log 7 and log 49 .
log 7 =ɺ 0.845098
log 49 =ɺ 1.690196
c) How would you express log 7 49 in terms of log 7 and log 49 ?
log 7 ÷ log 49 = log 7 49
2) a) Evaluate log 2 32 = 5
since 25= 32
b) Use your calculator to evaluate log 2 and log 32 .
log 2 =ɺ 0.301029996
log 32 =ɺ 1.505149978
c) How would you express log 2 32 in terms of log 2 and log 32 ?
log 32
log 2 32 =
log 2
3) How could we rewrite the general case, log a b (so you can use your calculator to find any log!)?
log b
log a b =
log a
MHF 4U Unit 7 - Laws of Exponents and Logarithms
Page 5 of 6.
Conclusion
Type
Exponent Law
Logarithm Law
Products
(a )(a )=
am+n
log a (mn ) = log a m + log a n
Quotients
a m ÷ a n = am−n
m
log a   = log a m − log a n
n
Powers
(a )
Zero exponents
a0 = 1
m
n
m n
= a m⋅n
log a 1 = 0
n
Negative exponents
a
−n
1
1
=   → n
a
a
m
log a m n = n ⋅ (log a m )
 1 
loga  m  = log a a − m
a 
= −m ⋅ log a a
in fact,
log a a = 1 therefore
 1 
loga  m  = − m
a 
= a m ⋅ bm
Power of a product
(ab)
Power of a quotient
 a m a m
  = m
b
b
m
Rational exponents
a n = n am
or
Identity
( a)
n
a
m
a1 = a
Change of base
log a a = 1
log a b =
log p b
log p a
, p ∈ℜ
however we mostly use p = 10 because this is what our
calculator has.
Assigned Work:
Pg. 466 #3abcd, 6
Pg. 475 #1ac, 2adef, 3acd, 4ac, 6bf, 8,0aef, 11acde
Pg. 466 #9
MHF 4U Unit 7 - Laws of Exponents and Logarithms
Page 6 of 6.