7.4 General Logarithmic and Exponential Functions:    

General Exponential Functions:

If a > 0 and r is any rational number, then,

                        a^r = (e^{ln a})^r = e^{r ln a}

therefore, even for irrational numbers x, we define: a^x = e^{x ln a}

Ex.  2 = e^{ln a} e^1.20 3.32

f(x) = a^x is called the exponential function with base a

lna^r = ln(e^{r ln a}) = rlna

thus, lna^r = rlna, for any real number r.

Laws of Exponents:

If x and y are real numbers and a, b > 0 then,

1. a^{x + y} = a^xa^y
2. a^{x – y} = a^x/a^y
3. (a^x)^y = a^xy
4. (ab)^x = a^xb^x

Differentiation Formula for Exponential Functions:

 (a^x) = a^xlna

Ex.

Exponential Integrals:

Ex.

The Power Rule:

If n is any real number and f(x) = xⁿ, then nx^n-1

3 Cases for Exponents and Bases:

1.       (a and b are constants)

2.  

3.  

Ex.  Differentiate

Use logarithmic differentiation –

Another method:

General Logarithmic Function:

  we see that

Cancellation Equations for the inverse functions  and  are and

For any positive number  we have,

Ex.  Evaluate

The Number e As a Limit:

because  we have,

therefore

7.5 Inverse Trigonometric Functions:

Definition of an Inverse:

  and 

Note:

Ex: evaluate

Cancellation Equations for Inverses:

 for

 for

 and

 for

 for

 and

Derivatives of Inverse Trig. Functions:

Examples/Exercises

Differentiate:

Differentiate:

Find:

Examples/Exercises/Web Links:

7.4

• http://en.wikipedia.org/wiki/Exponential_function
• http://www.2dcurves.com/exponential/exponentiale.html
• http://www.sosmath.com/algebra/logs/log4/log42/log42.html
• http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tutorials/unit3_3.html

7.5

• http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/invtrigderivdirectory/InvTrigDeriv.html
• http://archives.math.utk.edu/visual.calculus/3/inverse_trig.1/index.html
• http://www.acts.tinet.ie/inversetrig.functions_668.html
• http://www.mathematicshelpcentral.com/lecture_notes/precalculus_trigonometry_folder/inverse_trigonometric_functions.htm