1. Exponential and Logarithmic Functions
Chapter 9
Exponential and Logarithmic Functions

2. Chapter Sections 9.1 – Composite and Inverse Functions
9.2 – Exponential Functions
9.3 – Logarithmic Functions
9.4 – Properties of Logarithms
9.5 – Common Logarithms
9.6 – Exponential and Logarithmic Equations
9.7 – Natural Exponential and Natural Logarithmic Functions
Chapter 1 Outline

3. Natural Exponential and Natural Logarithmic Functions
§ 9.7
Natural Exponential and Natural Logarithmic Functions

4. The Natural Base, e
Both the natural exponential function and natural logarithmic function rely on an irrational number designated by the letter e.
The Natural Base, e
The natural base, e, is an irrational number that serves as the bases for the natural exponential function and the natural logarithmic function.

5. Identify the Natural Exponential Function
The natural exponential function is
where e is the natural base.

6. Identify the Natural Logarithmic Function
Natural Logarithms
Natural logarithms are logarithms with a base of e, the natural base. We indicate natural logarithms with the notation ln.
ln x is read “the natural logarithm of x”.

7. Identify the Natural Logarithmic Function
The natural logarithmic function is
where ln x = logex and e is the natural base.
Natural Logarithm in Exponential Form
For x > 0, if y = ln x, then ey = x.

8. Use the Change of Base Formula
For any logarithm bases a and b, and positive number x,

9. Solve Natural Logarithmic and Natural Exponential Equations
Properties for Natural Logarithms
Product Rule
Quotient Rule
Power Rule

10. Solve Natural Logarithmic and Natural Exponential Equations
Additional Properties for Natural Logarithms and Natural Exponential Expressions
Property 7
Property 8

11. Solve Natural Logarithmic and Natural Exponential Equations
Example Solve the equation ln y – ln (x + 9) = t for y.
Quotient Rule

12. Solve Applications Exponential Growth or Decay Formula
When a quantity P increases (grows) or decreases (decays) at an exponential rate, the value of P after time t can be found using the formula
where P0 is the initial starting value of the quantity P, and k is the constant growth rate or decay rate.
When k > 0, P increases as t increases.
When k < 0, P decreases and gets closer to 0 as t increases.

13. Solve Applications
Example When interest is compounded continuously, the balance, P, in the account at any time, t, can be calculate by the exponential growth formula. Suppose the interest rate is 6% compounded continuously and $1000 is initially invested. Determine the balance in the account after 3 years.
continued

14. Solve Applications
We are told that the principal initially invested, P0, is $ We are also given that the time, t, is 3 years and that the interest rate, k, is 6% or Substitute these values in the given formula.
After 3 years, the balance in the account is about $