Math 135
The Logarithm
Worksheet
• Rules of Logarithms
1. loga x = y ⇐⇒ ay = x
2. aloga M = M
3. loga a = 1
4. loga 1 = 0
5. loga M r = r loga M
6. loga (M · N ) = loga M + loga N
7. loga ( M
) = loga M − loga N
N
8. loga M =
logb M
logb a
• Common Mistakes
1. loga (M − N ) =
loga M
loga N
2. loga (M + N ) = loga (M N )
3.
logb M
logb a
=
M
a
• Arrange from least to greatest:
1. e, ln e, 12
2. e2 , 1, ln e2
3. ln 1e , e−1 , 1
4. 4, ln 4, e
• Simplify:
1. log10 10−3
5
2. log3 27 3
√
3. log2 2 8
4. (ln(e2 ))−1
5. 2log2 3 · 3log3 2
6. * log2 3 log3 4 log4 8
7. * eloge3 27
• Write as a sum or difference of logarithms without any exponents:
1. ln(x2 − y 2 )
2. log2
3. loga
4. logb
3x5
y8
q
5
2x
x2 −1
q
3
1
y2
·
University of Hawai‘i at Mānoa
q x2
z
161
R Spring - 2014
Math 135
The Logarithm
Worksheet
• Combine into a single logarithm:
1. log2 4x + log2 x + 2 log2 x
2.
1
[ln 2
3
4.
ln x3 +1
ln 2
+ ln y − ln y 2 − 4 ln y]
p
3. 31 loga x2 + loga x + y 2 − loga (x2 + y)
− log2 (x3 + 1) [Hint: Change of base.]
Sample Midterm
University of Hawai‘i at Mānoa
Sample Final
9 A B C D
13 A B C D
16 A B C D
23 A B C D
29 A B C D
40 A B C D
162
R Spring - 2014
Math 135
The Logarithm
Solutions
• Arrange from least to greatest:
1. e, ln e, 12
1
< ln e = 1 < e ≈ 2.718
2
2. e2 , 1, ln e2
1 < ln e2 = 2 < e2 ≈ 7.389
3. ln 1e , e−1 , 1
ln
1
= −1 < e−1 ≈ 0.368 < 1
e
4. 4, ln 4, e
ln 4 ≈ 1.386 < e ≈ 2.718 < 4
• Simplify:
1. log10 10−3 = −3
5
5
2. log3 27 3 = log3 (33 ) 3 = log3 35 = 5
√
√
1
5
3. log2 2 8 = log2 2 · 2 2 = log2 22+ 2 = log2 2 2 =
4. (ln(e2 ))−1 = 2−1 =
5
2
1
2
5. 2log2 3 · 3log3 2 = 3 · 2 = 6
6. log2 3 log3 4 log4 8
log2 3 log3 4 log4 8 = log2 3(log3 4 log4 8)
= (log3 4 log4 8) · log2 3
= log2 3(log3 4 log4 8)
log4 8
= log2 3log3 4
= log2 8
= 3
3
7. eloge3 27 = eloge3 3 = e3 loge3 3 = (e3 )
loge3 3
=3
• Write as a sum or difference of logarithms without any exponents:
1. ln(x2 − y 2 ) = ln(x + y)(x − y) = ln(x + y) + ln(x − y)
2.
log2
3x5
= log2 3x5 − log2 y 8
y8
= log2 3 + log2 x5 − log2 y 8
= log2 3 + 5 log2 x − 8 log2 y
3.
r
loga
5
University of Hawai‘i at Mānoa
2x
1
=
loga
2
x −1
5
2x
2
x −1
163
R Spring - 2014
Math 135
The Logarithm
Solutions
1
[loga 2x − loga (x2 − 1)]
5
1
=
[loga 2x − loga (x + 1)(x − 1)]
5
1
=
[loga 2 + loga x − (loga (x + 1) + loga (x − 1))]
5
1
1
1
1
=
loga 2 + loga x − loga (x + 1) − loga (x − 1)
5
5
5
5
=
4.
r
logb
3
1
·
y2
r
x2
z
!
=
1
1
x2
1
logb 2 + logb
3
y
2
z
1
1
logb y −2 + [logb x2 − logb z]
3
2
2
1
= − logb y + [2 logb x − logb z]
3
2
2
1
= − logb y + logb x − logb z
3
2
=
• Combine into a single logarithm:
1. log2 4x + log2 x + 2 log2 x = log2 4x + log2 x + log2 x2 = log2 (4x · x · x2 ) = log2 4x4
2.
1
1
[ln 2 + ln y − ln y 2 − 4 ln y] =
[ln 2 + ln y − (ln y 2 + 4 ln y)]
3
3
1
=
[ln 2y − (ln y 2 + ln y 4 )]
3
1
=
[ln 2y − ln y 6 ]
3 1
2y
=
ln
3
y6
r
2y
= ln 3 6
y
r
2
= ln 3 5
y
3.
p
p
2
1
loga x2 + loga x + y 2 − loga (x2 + y) = loga x 3 + loga x + y 2 − loga (x2 + y)
3
p
2
= loga (x 3 · x + y 2 ) − loga (x2 + y)
p
2
(x 3 · x + y 2 )
= loga
x2 + y
4.
ln x3 +1
ln 2
− log2 (x3 + 1) = log2 (x3 + 1) − log2 (x3 + 1) = log2 1 = 0
University of Hawai‘i at Mānoa
164
R Spring - 2014