Chapter 9 Exponential and Logarithmic Functions Copyright 2015

Chapter 9 Exponential and Logarithmic Functions Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -

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 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -2 2

§ 9. 7 Natural Exponential and Natural Logarithmic Functions Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -

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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -4 4

Identify the Natural Exponential Function The natural exponential function is where e is the natural base. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -5 5

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”. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -6 6

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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -7 7

Use the Change of Base Formula For any logarithm bases a and b, and positive number x, Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -8 8

Solve Natural Logarithmic and Natural Exponential Equations Properties for Natural Logarithms Product Rule Quotient Rule Power Rule Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -9 9

Solve Natural Logarithmic and Natural Exponential Equations Additional Properties for Natural Logarithms and Natural Exponential Expressions Property 7 Property 8 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -10 10

Solve Natural Logarithmic and Natural Exponential Equations Example Solve the equation ln y – ln (x + 9) = t for y. Quotient Rule Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -11 11

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 P 0 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -12 12

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 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -13 13

Solve Applications We are told that the principal initially invested, P 0, 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. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 9 -14 14