Chapter 5 Outline: Exponential and Logarithmic Functions
5.1 Exponential Functions
1. Graph the function: 𝑦𝑦 = 𝑏𝑏 𝑥𝑥
a) Domain: (−∞, ∞)
b) Range: (0, ∞)
c) y-intercept: (0,1)
d) Asymptote: 𝑦𝑦 = 0
2. Reflections
a) Reflect 𝑦𝑦 = 𝑏𝑏 𝑥𝑥 about the 𝑥𝑥-axis:
𝑦𝑦 = −𝑏𝑏 𝑥𝑥
𝑥𝑥
𝑦𝑦 = 𝑏𝑏 −𝑥𝑥
b) Reflect 𝑦𝑦 = 𝑏𝑏 about the 𝑦𝑦-axis:
c) Reflect 𝑦𝑦 = 𝑏𝑏 𝑥𝑥 about the origin:
𝑦𝑦 = −𝑏𝑏 −𝑥𝑥
3. Understand how the magnitude of 𝑏𝑏 affects the shape.
4. Understand how 𝑏𝑏 > 1 or 0 < 𝑏𝑏 < 1 affects the shape.
5.2 The Exponential Function: 𝒚𝒚 = 𝒆𝒆𝒙𝒙
1. 𝑒𝑒 is an irrational number which is approximately 2.718, which is the base that has a tangent
line slope of 1 at (0,1).
5.3 Logarithmic Functions
Assume 𝑏𝑏 is positive.
1. Graph the function: 𝑦𝑦 = log 𝑏𝑏 (𝑥𝑥)
a) Domain: (0, ∞)
b) Range: (−∞, ∞)
c) Point: (1,0)
d) Asymptote: 𝑥𝑥 = 0
2. Reflections
a) Reflect 𝑦𝑦 = log 𝑏𝑏 (𝑥𝑥) about the 𝑥𝑥-axis: 𝑦𝑦 = − log 𝑏𝑏 (𝑥𝑥)
b) Reflect 𝑦𝑦 = log 𝑏𝑏 (𝑥𝑥) about the 𝑦𝑦-axis: 𝑦𝑦 = log 𝑏𝑏 (−𝑥𝑥)
c) Reflect 𝑦𝑦 = log 𝑏𝑏 (𝑥𝑥) about the origin: 𝑦𝑦 = − log 𝑏𝑏 (−𝑥𝑥)
3. 𝑦𝑦 = 𝑏𝑏 𝑥𝑥 and 𝑦𝑦 = log 𝑏𝑏 (𝑥𝑥) are inverse functions of each other, which implies:
a) 𝑏𝑏 log𝑏𝑏(𝐴𝐴) = 𝐴𝐴
b) log 𝑏𝑏 (𝑏𝑏 𝐴𝐴 ) = 𝐴𝐴
𝑥𝑥
4. 𝑦𝑦 = 𝑒𝑒 and 𝑦𝑦 = ln(𝑥𝑥) are inverse functions of each other, which implies:
a) 𝑒𝑒 ln(𝐴𝐴) = 𝐴𝐴
b) ln(𝑒𝑒 𝐴𝐴 ) = 𝐴𝐴
5. Equivalent log statements
a) 𝑦𝑦 = log 𝑏𝑏 (𝑥𝑥) is equivalent to 𝑏𝑏 𝑦𝑦 = 𝑥𝑥.
b) 𝑦𝑦 = ln(𝑥𝑥) is equivalent to 𝑒𝑒 𝑦𝑦 = 𝑥𝑥.
5.4 Properties of Logarithms
Assume 𝑏𝑏 is positive.
1. log 𝑏𝑏 (𝑏𝑏) = 1
2. log 𝑏𝑏 (1) = 0
3. log 𝑏𝑏 (𝐴𝐴𝐴𝐴) = log 𝑏𝑏 (𝐴𝐴) + log 𝑏𝑏 (𝐵𝐵)
𝐴𝐴
4. log 𝑏𝑏 � � = log 𝑏𝑏 (𝐴𝐴) − log 𝑏𝑏 (𝐵𝐵)
𝐵𝐵
for 𝐴𝐴, 𝐵𝐵 > 0
for 𝐴𝐴, 𝐵𝐵 > 0
5. log 𝑏𝑏 (𝐴𝐴𝐵𝐵 ) = 𝐵𝐵 log 𝑏𝑏 𝐴𝐴
for 𝐴𝐴 > 0
6. Change of base formula from log base 𝑎𝑎 to log base 𝑏𝑏.
log 𝑏𝑏 (𝑥𝑥)
log 𝑎𝑎 (𝑥𝑥) =
log 𝑏𝑏 (𝑎𝑎)
5.5 Equations and Inequalities with Logs and Exponents
𝑚𝑚 log 𝑏𝑏 |𝑥𝑥| 𝑚𝑚 is an even integer
𝑚𝑚 log 𝑏𝑏 (𝑥𝑥) 𝑚𝑚 is an odd integer
𝑚𝑚
log 𝑏𝑏 |𝑥𝑥| 𝑚𝑚 is an even integer
𝑛𝑛
2. log 𝑏𝑏 (𝑥𝑥 𝑚𝑚/𝑛𝑛 ) = � 𝑚𝑚
log 𝑏𝑏 (𝑥𝑥) 𝑚𝑚 is an odd integer
1. log 𝑏𝑏 (𝑥𝑥 𝑚𝑚 ) = �
𝑚𝑚
𝑛𝑛
Assume is a rational number reduced to lowest terms.
𝑛𝑛
3. Because of the domain restriction for logarithms, solving equations involving logs often
requires checking for extraneous solutions.
4. If 𝑓𝑓(𝑥𝑥) is an increasing and continuous function and 𝑎𝑎 < 𝑏𝑏 , then 𝑓𝑓(𝑎𝑎) < 𝑓𝑓(𝑏𝑏).
a) Let 𝑓𝑓(𝑥𝑥) = 𝑒𝑒 𝑥𝑥 .
If 𝑎𝑎 < 𝑏𝑏,
then 𝑒𝑒 𝑎𝑎 < 𝑒𝑒 𝑏𝑏 .
b) Let 𝑓𝑓(𝑥𝑥) = ln(𝑥𝑥).
If 0 < 𝑎𝑎 < 𝑏𝑏,
then ln(𝑎𝑎) < ln(𝑏𝑏).
Examples:
c) Let 𝑓𝑓(𝑥𝑥) = √𝑥𝑥.
If 0 ≤ 𝑎𝑎 < 𝑏𝑏,
then √𝑎𝑎 < √𝑏𝑏.
d) Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 3 .
If 𝑎𝑎 < 𝑏𝑏,
then 𝑎𝑎3 < 𝑏𝑏 3 .
e) Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 , 𝑥𝑥 ≥ 0.
If 0 < 𝑎𝑎 < 𝑏𝑏,
then 𝑎𝑎2 < 𝑏𝑏 2 .
5. If 𝑓𝑓(𝑥𝑥) is a decreasing and continuous function and 𝑎𝑎 < 𝑏𝑏, then 𝑓𝑓(𝑎𝑎) > 𝑓𝑓(𝑏𝑏).
a) Let 𝑓𝑓(𝑥𝑥) = −𝑥𝑥
If 𝑎𝑎 < 𝑏𝑏,
then −𝑎𝑎 > −𝑏𝑏
1
1
1
b) Let 𝑓𝑓(𝑥𝑥) = , 𝑥𝑥 ≥ 0.
If 0 < 𝑎𝑎 < 𝑏𝑏,
then > .
𝑥𝑥
1
c) Let 𝑓𝑓(𝑥𝑥) = , 𝑥𝑥 ≤ 0.
𝑥𝑥
d) Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 , 𝑥𝑥 ≤ 0
If 𝑎𝑎 < 𝑏𝑏 < 0,
If 𝑎𝑎 < 𝑏𝑏 < 0,
𝑎𝑎
1
𝑏𝑏
1
then > .
𝑎𝑎
𝑏𝑏
then 𝑎𝑎2 > 𝑏𝑏 2 .