8.6 Natural Logarithms

Learning Objective: After this lesson, I will be able to solve equations using
natural logs.

Language Objective: After this lesson, I will be able to explain how to switch
between exponential functions with base e and natural logs.
What is “e” ?
e ≈ 2.71828
Try these on your calculator:
1. e5
2. 2e3
Natural logarithmic function:
3. e-1
4.
1
e
loge x = y is commonly written as ln x = y
Y = ex and y = ln x are inverses.
Simplifying Natural Logs: You can use the same product, quotient, and power
rules with natural logs as with other logs.
Examples: Write as a single log.
1. 3 ln 6 – ln 8
2. 3 ln x + ln y
Examples: Expand.
x
1. ln  
 y
2
2. ln a3 b
How do I solve a natural log equation? First, change it to exponential form
Examples: Solve for x
1. ln (3x + 5)2 = 4
2. ln x = 0.1
3. ln (3x - 9) = 21
4. ln (
x2
) = 12
3
Solving an Exponential Equation with e : You can use natural logs!
Examples: Use natural logs to solve each equation:
2x
1. 7e2x + 2.5 = 20
2. ex+1 = 30
p. 481 #1-9 odd, 10, 14-22 even, 23-28 all
3. e 5  7.2  9.1