6.7 – Base “e” and Natural Logarithms Objective – TSW solve equations that contain “e” and natural logarithms.

What is this “e” business about ? “e” is a number like pi. - It is a non repeating, never ending decimal. - An irrational number. - e ≈ 2.71828… - e is the base of the natural log (ln)…just like 10 is the base of log.

On your calculators… Let’s Practice Using “e” e9 2. e3.4 3. 3e0.05 4. e1/4

Why do we use “e” Because in the natural log (ln) the base is e. ALL PROPERTIES of logarithms apply to natural logarithms. Anytime you see ln…you can write loge  For example: ln 3 means loge 3

Let’s look at this “ln” business. ln 99,999 2. ln 0.994 3. ln ½ 4. ln (-2) 5. ln (15)

Examples…Write each exponential equation in logarithmic form or each logarithmic equation in exponential form. ex = 8 2. e5 = x 3. e7 = x 4. ln x = 0.7741 5. ln 10 = x 6. ln x = 2.1438

Write each expression as a single logarithm. 3 ln 10 – ln 8 8. ln 40 + 2ln ½ + ln x 9. 6ln 8 – 2ln 4 10. 2ln 5 + 4ln 2 + ln 5y

To Solve natural log equations: Identify whether the equation is in log form or exponential form. Get the “log/ln” portion by itself OR get the “exponential” portion by itself. Rewrite into either log or exponential form. Solve. Check your answer. ***REMEMBER ln means loge***

11. ex+2 = 7 12. e3x + 1= 5

13. 3 ln x – 1 = 4 14. 2ln x + 2 = 6

Continuous Compound Interest

Homework… Pg. 529 #’s 1-14(all), 34, 36, 38 