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Section 1.4: Exponential and Logarithmic Functions
In this section, we review the basics of exponential and logarithmic functions. We first
give a brief review of the laws of exponents.
Properties of Exponents
Let a > 0 and b > 0. Then the following properties hold
1. a 0  1
5.
ax
a
2. a a  a
x y
x y
3. (a x ) y  a xy
y
 a x y
x
ax
a
6.    x
b
b
7. a  x 
1
ax
4. (ab) x  a x b x
Example 1: Use the properties of exponents to simplify the following expressions.
a. 2 4
b. (33 )(34 )
 e5 
c.  2 
e 
 
1
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Fractional Exponents
 
Recall that a1 / n  n a and a m / n  a m
Note:
1/ n
 n am .
a  2 a  (a)1/ 2
Example 2: Simplify the following.
a. 91 / 2
b. 8 2 / 3
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Exponential Functions
The exponential function with base a is denoted by
f ( x)  a x
where a > 0, a  1 , and x is any real number.
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Graphs of Exponential Functions
Example 3: Graph f ( x)  2 x and g ( x)  3 x .
Solution:
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Example 4: Graph f ( x)  2  x and g ( x)  3 x .
Solution: Using the following point plot tables, we produce the graphs to the right.
x
y  f ( x)  2  x
x
y  g ( x)  3 x
-2
-1
0
1
2 (2)  2 2  4
2 (1)  21  2
20  1
1
2 1 
2
1
1
2 2  2 
4
2
-2
3  ( 2 )  3 2  9
-1
3 ( 1)  31  3
0
30  1
1
3 1 
3
1 1
3 2  2 
9
3
2
1
2
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In general,
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The Number e
The number e is an irrational number approximated by
e  2.7182818
The function f ( x)  e x is called the exponential function of base e.
Example 5: Determine the number e, e 2 , and e 3 on a calculator.
Solution:
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Example 6: Graph f ( x)  e x and g ( x)  e  x on the same graph.
Solution: The following displays the graphs of these two functions on the same graph:
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Solving Exponential Equations with Like Bases
Uses fact that if
a x  a y , then x  y .
Example 7: Solve 25 x  32 for x.
Solution:
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Example 8: Solve e 5 x  e for x.
Solution: If we note that e  e1 , then the e 5 x  e becomes e 5 x  e1 . Since the bases are
the same (the base e) on both sides of the equation, we can equate the exponents and solve
for x. That is,
e 5 x  e1
5x  1
1
x
5
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Natural Logarithm Function
Given by
y  ln x  log e x
Note: log e x  y means e y  x .
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Example 9: Write the equation ln 3  1.0986... as an exponential equation.
Solution: Using the fact that ln x  log e x  y means e y  x , we convert as follows:
ln 3  1.0986...
log e 3  1.0986...
e1.0986...  3
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Example 10: Write the equation e 2  7.389... as an logarithmic equation.
Solution:
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Calculating The Natural Logarithm on a Calculator
Example 11: Determine ln 7 , ln 0.4 , ln 0 , and ln - 3 on a calculator.
Solution:
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Properties of Natural Logarithms
1. ln 1 
2. ln e 
3. ln x  0 if x > 1.
4. ln x  0 if 0 < x < 1.
5. ln x is undefined if x  0 .
6. ln e x  x, e ln x  x
(Inverse Properties)
Example 12: Apply the inverse properties of ln x and e x to simplify ln e x and e ln( 4 x3) .
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Solution:
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Additional Logarithmic Properties – Natural Logarithm of a product,
Quotient, and Exponent Laws
7. ln uv  ln u  ln v
8. ln
u
 ln u  ln v
v
9. ln u k  k ln u
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Example 13: Use the properties of logarithms to simplify ln
e
.
5
Solution: On this, we perform the following steps:
ln
e
 ln e  ln 5
5
(Use quotient property 8 , that is, ln
 1  ln 5
(Recall ln e  1)
 0.6904
(Calculato r Approximat ion)
u
 ln u  ln v)
v
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Example 14: Use the properties of logarithms to expand the expression ln( x x 3  1 ) .
Solution:
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Example 15: Write the expression 5 ln x  2 ln y  3 ln z as the logarithm of a single
quantity.
Solution:
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Graph of the Natural Logarithm Function
Example 16: Graph f ( x)  ln x
Solution:
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Solving Equations with Natural Logarithms
Idea is to isolate the natural logarithm by itself on one side of the equation and use the
definition of the natural logarithm to convert to an exponential equation.
Example 15: Solve 2 ln x  4 for x.
Solution: We solve this equation using the following steps:
2 ln x  4
4
ln x 
2
ln x  2
log e x  2
x  e 2  7.389
(Divide both sides by 2 to isolate ln term)
(Simplify)
(Recall by definition ln x  log e x)
(Convert t o exponentia l form using fact log e x  y means x  e y )
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Solving Exponential Equations with Unlike Bases using Logarithms
Steps
1. Isolate the exponential term by itself on one side of the equation.
2. Take the natural logarithm of both sides and use the property of logarithms ln u k  k ln u
to simplify.
3. Solve the resulting equation for x.
Example 17: Solve e 0.5 x  2 for x.
Solution:
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Example 18: Solve 2e x1  5  9 for x.
Solution: Before taking the natural logarithm of both sides, the exponential e needs to be
isolated on one side of the equation. We solve using the following steps:
2e x 1  5  9
2e x 1  9  5
(Add 5 to both sides)
2e x 1  14
(Simplify)
e x 1 
14
2
(Divide both sides by 2)
e x 1  7
(Simplify)
ln e x 1  ln 7
(Take ln of both sides)
( x  1) ln e  ln 7
( x  1)(1)  ln 7
(Bring exponent down using property ln u k  k ln u )
(Recall ln e  1)
x  1  ln 7
x  1  ln 7  2.9459
(Simplify )
(Solve for x)
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