Define exponential functions that have bases other than eDifferentiate and integrate functions that have other basesUse 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:1.a0 = 12.axay = ax+y

3.ax/ay = ax – y

4.(ax)y= axy

Ex 1 p. 360 Radioactive Half-life ModelThe 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

57151

2

t

y

If t = 0, y = 1 gram. If t = 5715, then y = ½, which would be correct.

1000057151

.29734682

y

Remember, this is just the change of base rule you’ve seen before, just in a new setting!

Logarithmic Properties still apply

a

a a a

a a a

a

1. log 1 0

2. log log log

3. log log log

4. log logna

xy x y

xx y

y

x n x

Exponential functions and logarithmic functions are inverse functions

( ) xf x a( ) logag x x

So as review, we’ll work with these properties with bases other than base eEx. 2a, p. 361

13

243x Take the log base 3 to each side

3 3

1log 3 log

243x

53log 3x

5x

Ex. 2b, p. 361

27 log x Exponentiate each side using base 2

2log72 2 x72 x

128 x

When thinking of these derivatives, it is often helpful to think of them as natural exponential things or as natural log things.

lna xxd da e

dx dx

ln lna x de a x

dx

ln ln lna x xe a a a

logad

xdx

1ln

ln

dx

dx a

1 1

ln a x

Ex 3 p. 362 Differentiating functions to other bases.Find the derivative of each function.

. 2xa y ' (ln 2)2xy 5. 2 xb y 5' (ln 2)2 (5)xy 55ln 2 2 x

. logsinc y x log with no base shown is a common log, base 10cos

'(ln10)sin

xy

x cot

ln10

x

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:

(ln )x a xa e1

lnx xa dx a C

a

Ex 4 p. 363 Integrating an exponential function with base 33xdx

13

ln 3x C

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!

2d.

dxea e 0 Constant rule

2d.

dxxb e 22 Exponential rulexe

d.

dxec x

1 Power ruleeex

. xdd xdx

xy x Logarithmic differentiation

required!ln ln xy x lnx x' 1

ln (1)y

x xy x

1 ln x ' 1 lny y x 1 lnxx x

5.5a p.366/ 3-60 mult 3