# Unit 7: Logarithms and Exponents

The natural logarithm is considered one of the most beautiful things about mathematics. Euler's number, the base of natural logarithms, arises in a mysterious way in many things (hence the name ``natural''. Logs and exponents allow us to do things that we weren't able to do before with calculus.

Many students do not have a strong background with logs and exponents. You are enouraged to look over the basic rules of logs and exponents before diving into the calculus.

Also covered in this unit are hyperbolic trig functions. Hyperbolic trig funtions are defined using exponents, yet have behaviors comporable to normal trig functions.

### Objectives:

After compelting this unit, you should:
• Understand how the value of Euler's number, e, was derived.
• Be able to differentiate logs and exponents, as well composite functions involving logs and exponents.
• Be able to do problems involving exponential growth and decay, such as radioactive decay and population growth problems.
• Understand the derivatives of hyperbolic trig functions.

### Suggested procedure:

• Read Simmons Chapter 8 and section 9.7.
• Read the following World Web Math pages (currently being revised):
• Euler's Magic Number
• Differentiation of logarithms
• Derivatives of Hyperoblic Trigonometric functions
• Simmons does not cover hyperbolic trig functions. Ask your instructor for some material on this topic.
• Do some problems in Simmons:
• 8.2 : 4, 58
• 8.3 : 1, 5, 11, 16, 17, 25
• 8.4 : 1, 4, 5, 20, 21
• 8.5 : 11, 13, 16
• 8.6 : 5, 6, 7, 8
• Take the Practice Test, Xdvi or PDF.