CHAPTER 5 SECTION 5 5 BASES OTHER THAN

CHAPTER 5 SECTION 5. 5 BASES OTHER THAN e AND APPLICATIONS

Definition of Exponential Function to Base a

0 • a =1 x y x+y • a a = a x x-y • a = a y a x y xy • (a ) = a

Definition of Logarithmic Function to Base a

Properties of Inverse Functions

Theorem 5. 13 Derivatives for Bases Other Than e

Examples for Derivatives for Bases other than e

Theorem 5. 14 The Power Rule for Real Exponents

Differentiate. Hint: Take the ln of both sides.

Find each derivative with respect to the given variable.

LOGARITHMIC DIFFERENTIATION • Differentiate . – Using logarithmic differentiation, we have:

To integrate exponential functions other than base e, either • Convert to base e using the formula Or q Integrate directly using the integration formula

EXP. FUNCTIONS WITH BASE a • Example 14

Theorem: Proof:

Theorem 5. 15 A Limit Involving e

Applications of Exponential Functions • Compound Interest Formulas – Compounded n times per year: – Compounded continuously: Where t = number of years, P = Initial deposit, A = balance after t years, r = interest rate expresses as a decimal, n = number of compounding periods per years.

A deposit of $2500 invested into an account paying an interest rate of 10%. Find its balance after 5 years if interest is compounded… a. quarterly b. monthly c. continuously