1. 7.7 – Base e and Natural Logarithms

2. The number e.
The history of mathematics is marked by the discovery of the special numbers such as zero, π and i. Another number, like π and i, that is denoted by a letter is the number e.

3. Natural Logarithm Natural Logarithm: log with base e: loge x
Notation: ln x
e is the base e is the base
x is the argument y is the exponent

4. Example 1: Write in logarithmic form.
A. ey = 23 B. e4 = x

5. Example 2: Write in exponential form.
A. ln 25 = y
B. ln x = 47

6. Example 3a: Solve Base e Equations
Round to the nearest thousandth. 3e–2x + 4 = 10.

7. Example 3b:
Solve the exponential equation. Round to the nearest thousandth. 6e3x + 5 = 15

8. Example 4: Condensing
A. Write 4 ln 3 + ln 6 as a single logarithm. B. Write 2 ln 3 + ln 4 - ln y as a single logarithm.

9. Example 5a: Solve Natural Log Equations
Solve. Round to the nearest thousandth.
ln 5x + 3 = 6

10. Example 5b: Solve Natural Log Equations
Solve. Round to the nearest thousandth.
3 ln (6x – 2) = 12

11. Example 5c: Solve Natural Log Equations
Solve. Round to three decimals.
3 ln ln x = 6

12. Example 5d: Solve Natural Log Equations
Solve. Round to three decimals.
ln (x + 5) = ln (x – 1) – ln (x + 1)

13. Example 6a: Solve Natural Log Inequalities
Solve. Round to the nearest thousandth.
ln (4x – 2) < 7

14. Example 6b: Solve Natural Log Inequalities
Solve. Round to the nearest thousandth.
ln (3x + 1)2 > 8

16. Example 7:
Suppose you deposit $700 into an account paying 3% annual interest, compounded continuously. What is the balance after 8 years?

17. Example 8:
Suppose you deposit $700 into an account paying 2.4% annual interest, compounded continuously. How long will it take for the balance in your account to reach at least $1200?