Define exponential functions that have bases other than e Differentiate and integrate functions that have other bases Use Exponential functions to model compound interest and exponential growth
The base of the natural exponential function is e . This base can be used to assign meaning to a general base a
The laws of exponents apply here as well:
a 0 = 1
a x a y = a x+y
a x /a y = a x – y
(a x ) y = a xy
Ex 1 p. 360 Radioactive Half-life Model The half-life of carbon-14 is about 5715 years. A sample contains 1 gram of carbon-14. How much will be present in 10,000 years? Solution: Let t = 0 represent the present time and y represent the amount (in grams) of carbon-14. Using a base of ½, you can model y by the equation If t = 0, y = 1 gram. If t = 5715, then y = ½, which would be correct.
Remember, this is just the change of base rule you’ve seen before, just in a new setting!
Logarithmic Properties still apply Exponential functions and logarithmic functions are inverse functions
So as review, we’ll work with these properties with bases other than base e Ex. 2a, p. 361 Take the log base 3 to each side
Ex. 2b, p. 361 Exponentiate each side using base 2
When thinking of these derivatives, it is often helpful to think of them as natural exponential things or as natural log things.
Ex 3 p. 362 Differentiating functions to other bases. Find the derivative of each function. log with no base shown is a common log, base 10
Sometimes an integrand will work with a exponential function involving another base than e. When this occurs, we can do one of two things – convert to base e using the formula and integrate, or integrate directly, using the following formula: Ex 4 p. 363 Integrating an exponential function with base 3
When the power rule was introduced in Ch. 2, we limited it to rational exponents. Now the rule is extended to cover all real exponents.
This next example compares the derivatives of four different functions involving exponents. Be CAREFUL! Logarithmic differentiation required!