An exponential function with the base b is the function of the form f(x) = 𝑏 𝑥 or y = 𝑏 𝑥 , where
b > 0, b ≠ 1.
Tell whether each statement is True or False.
1. f(x) = 3𝑥 2 is an exponential function. False
2. if f(x) = 2𝑥 , then f(-1) 2. True
3. The function f(x) = 1𝑥 is an exponential function. False
4. The function f(x) = (−3)𝑥 is an exponential function. False
1
5. the functions 𝑓(𝑥) = (3)𝑥 and g(x) = 3−𝑥 have the same graph True
Complete the table of values.
X -3 -2 -1 0 1 2 3
y = (𝟑)𝒙
𝟏 27 9 3 1 1 1 1
3 9 27
y = 𝟏𝟎𝒙 1 1 1 1 10 100 1000
1000 100 10
y = 𝟎. 𝟖𝒙 1 1 1 1 0.8 0.64 0.512
0.512 0.64 0.8
A. Evaluate the function f(x) = 3𝑥 at
1
a.) f(2) b.) f(2) c.) f(0.4)
B. Tell whether the given is an exponential function or not.
1. 𝑓(𝑥) = 2𝑥 2
2. 𝑓(𝑥) = −2𝑥 2
3. 𝑓(𝑥) = 12𝑥
4. 𝑓(𝑥) = 2𝑥
5. 𝑓(𝑥) = 2−𝑥
1
6. 𝑓(𝑥) = (3)𝑥+4
Describe the following expressions:
1. y = 2𝑥
2.4𝑥−1 = 16𝑥
3. 2𝑥 ≥ 26
Exponential Equation
An equation involving exponential expression
Example: 4𝑥−1 = 16𝑥
Exponential Inequality
An inequality involving exponential expression
Example: 2𝑥 ≥ 26
Exponential Function
Function of the form f(x) = 𝑏 𝑥 or y = 𝑏 𝑥 , where b > 0, b ≠ 1
Example: y = 2𝑥
Property of Exponential Inequality
If b > 1, then the exponential function y = 𝑏 𝑥 is increasing for all x. This means that
𝑏 𝑥 < 𝑏 𝑦 if and if x < y.
If 0 < b < 1, then the exponential function y = 𝑏 𝑥 is decreasing for all x. This means
that 𝑏 𝑥 > 𝑏 𝑦 if and only if x > y.
The resulting direction of the inequality (m < n or m > n) is based on whether the
base b is greater than 1 or less than 1.
Find the base of the exponential function whose graph contains the given points.
a. (1,4)
b. (2/3, 4)
a. y = 𝑏 𝑥 b. y = 𝑏 𝑥
3
4 = 𝑏1 42 = b
b=4 23 = 𝑏
8=b
Review exponents by finding the value of x for the given equation.
a.24 = x 16
b. 43 = x 64
𝟏
c.5−1 = x 𝟓
1
𝟏
d. 16−2 = x 𝟒
Logarithmic Form
𝑙𝑜𝑔𝑏 𝑎 = 𝑐
Exponential Function
𝑏𝑎 = 𝑎
Find the value of the following logarithmic expressions.
1. 𝑙𝑜𝑔2 32 5
2. 𝑙𝑜𝑔9 729 3
3. 𝑙𝑜𝑔5 5 1
4. 𝑙𝑜𝑔1/2 16 -4
5. 𝑙𝑜𝑔1 1 0
𝟏
6. 𝑙𝑜𝑔1 /√5 -𝟐
Rewrite the following exponential equations in logarithmic form whenever possible.
a. 53 = 125
1
b. 7−2 =
49
c. 102 = 100
22 4
d. =
3 9
e. 0.1− 4 = 10,000
f. −22 = 4
g. 𝑒 2 = x
h. 76 = 21
i. 4 0 = 1
Describe the following logarithmic expressions.
a. 𝑙𝑜𝑔3 (𝑥 − 2) = 5
b. 𝑙𝑜𝑔5 x < 2
c. y = 𝑙𝑜𝑔1/2 𝑥
Logarithmic Equation
An equation involving logarithm
Example: 𝑙𝑜𝑔3 (𝑥 − 2) = 5
Logarithmic Inequality
An inequality involving logarithm
Example: 𝑙𝑜𝑔5 x < 2
Logarithmic Function
A function defined by y = 𝑙𝑜𝑔𝑏 𝑥, if and only if x = 𝑏 𝑦 where x and b are positive real numbers,
and 𝑏 ≠ 1.
Example: y = 𝑙𝑜𝑔1/2 𝑥
Determine whether the given is a logarithmic function, logarithmic equation, logarithmic
inequality or none of these.
1. 𝑙𝑜𝑔𝑛 36 = 2
2. f(x) = 2𝑥
3. y = 3 𝑙𝑜𝑔2 𝑦
4. . 𝑙𝑜𝑔3 (2𝑥 − 1) > 𝑙𝑜𝑔3 (𝑥 + 2)
5. 𝑙𝑜𝑔10 10,000 = 4
6. ln x > 1
a. 𝑙𝑜𝑔𝑏 𝑏 = 1
b. 𝑙𝑜𝑔𝑏 1 = 0
c. 𝑙𝑜𝑔𝑏 (𝑏 𝑥 ) = 𝑥 (𝑏 > 0, 𝑏 ≠ 1)
a. 𝑙𝑜𝑔𝑏 (𝑢𝑣) = 𝑙𝑜𝑔𝑏 𝑢 + 𝑙𝑜𝑔𝑏 𝑣
𝑢
b. 𝑙𝑜𝑔𝑏 (𝑣 ) = 𝑙𝑜𝑔𝑏 𝑢 − 𝑙𝑜𝑔𝑏 𝑣
c. 𝑙𝑜𝑔𝑏 (𝑢𝑛 ) = 𝑛 𝑙𝑜𝑔𝑏 𝑢
Example:
a. 𝑙𝑜𝑔 (𝑎𝑏 2 )
3
b. 𝑙𝑜𝑔3 (𝑥) 3
c. ln [ x (x – 5)]
Some situations can only be described by more than one formula, depending on the value of the
independent variable.
Example:
1. A jeepney ride cost 8.00 pesos for the first 4 kilometer.
f(d) 8 0<d≤4
8 = 1.5 d d>4
(𝑥 𝑎 ) (𝑥 𝑏 ) = (𝑥 𝑎+𝑏 )
(𝑥 𝑎 ) 𝑏
= 𝑥 𝑎𝑏
(𝑥𝑦)𝑎 = 𝑥 𝑎 𝑦 𝑎
(𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2
(𝑥 − 𝑦)2 = 𝑥 2 − 2𝑥𝑦 + 𝑦 2
(𝑥 + 𝑦)3 = 𝑥 3 + 3𝑥 2 𝑦 + 3𝑥𝑦 2 + 𝑦 3
(𝑥 − 𝑦)3 = 𝑥 3 − 3𝑥 2 𝑦 + 3𝑥𝑦 2 − 𝑦 3
(x + y) (x – y) = 𝑥 2 − 𝑦 2
(x + y + z)2 = 𝑥 2 + 𝑦 2 + 𝑧 2 + 2𝑥𝑦 + 2𝑥𝑧 + 2𝑦𝑧
