1. 8-6 Natural Logarithms p. 462 Obj: To be able to solve equations using natural logarithms.

2. Natural Log Function If, then, which is commonly written as.

3. 5 ln 2 – ln 4 Simplifying Natural Logarithms Example 1 Write each expression as a single natural logarithm. = ln 2 5 – ln 4 ¼ ln 3 + ¼ ln x = ln 8 a. b. = ln (32/4) = ln 3 ¼ + ln x ¼ = ln 3 ¼ ∙x ¼ = ln (3x) ¼

4. Natural Logarithm y = e x ln y = x INVERSE When solving a natural logarithmic equation, rewrite each side of the equation with e as the base to “undo” ln!

5. Solving a Natural Log Equation Example 2 Solve.

6. Solving a Natural Log Equation Example 3 Solve.

7. So, what is the difference between logs and natural logs? The big concept to take away from this section is using e when solving natural logarithmic equations. The same log rules apply for natural logs!

8. Ticket Out Use natural logarithms to solve e –6x = 3.1. Solve ln 2 + ln x = 5. Write 3 ln a – ½ (ln b + ln c 2 ) as a single natural logarithm. Simplify ln e –4.

9. Homework p. 464 #2-30 even