Exponential and Logarithmic Functions Chapter 9 Exponential and Logarithmic Functions

Chapter Sections 9.1 – Composite and Inverse Functions 9.2 – Exponential Functions 9.3 – Logarithmic Functions 9.4 – Properties of Logarithms 9.5 – Common Logarithms 9.6 – Exponential and Logarithmic Equations 9.7 – Natural Exponential and Natural Logarithmic Functions Chapter 1 Outline

Natural Exponential and Natural Logarithmic Functions § 9.7 Natural Exponential and Natural Logarithmic Functions

The Natural Base, e Both the natural exponential function and natural logarithmic function rely on an irrational number designated by the letter e. The Natural Base, e The natural base, e, is an irrational number that serves as the bases for the natural exponential function and the natural logarithmic function.

Identify the Natural Exponential Function The natural exponential function is where e is the natural base.

Identify the Natural Logarithmic Function Natural Logarithms Natural logarithms are logarithms with a base of e, the natural base. We indicate natural logarithms with the notation ln. ln x is read “the natural logarithm of x”.

Identify the Natural Logarithmic Function The natural logarithmic function is where ln x = logex and e is the natural base. Natural Logarithm in Exponential Form For x > 0, if y = ln x, then ey = x.

Use the Change of Base Formula For any logarithm bases a and b, and positive number x,

Solve Natural Logarithmic and Natural Exponential Equations Properties for Natural Logarithms Product Rule Quotient Rule Power Rule

Solve Natural Logarithmic and Natural Exponential Equations Additional Properties for Natural Logarithms and Natural Exponential Expressions Property 7 Property 8

Solve Natural Logarithmic and Natural Exponential Equations Example Solve the equation ln y – ln (x + 9) = t for y. Quotient Rule

Solve Applications Exponential Growth or Decay Formula When a quantity P increases (grows) or decreases (decays) at an exponential rate, the value of P after time t can be found using the formula where P0 is the initial starting value of the quantity P, and k is the constant growth rate or decay rate. When k > 0, P increases as t increases. When k < 0, P decreases and gets closer to 0 as t increases.

Solve Applications Example When interest is compounded continuously, the balance, P, in the account at any time, t, can be calculate by the exponential growth formula. Suppose the interest rate is 6% compounded continuously and $1000 is initially invested. Determine the balance in the account after 3 years. continued

Solve Applications We are told that the principal initially invested, P0, is $1000. We are also given that the time, t, is 3 years and that the interest rate, k, is 6% or 0.06. Substitute these values in the given formula. After 3 years, the balance in the account is about $1197.22.