The essence of mathematics is not to make simple things complicated, but to make complicated things simple. Stan Gudder John Evans Professor of Mathematics University of Denver

Chapter 3 Exponential and Logarithmic Functions

Day I Exponential and Logarithmic Equations (3.1)

Exponential functions can be used to model the amount of defoliation caused by the gypsy moth.

GOAL I. To recognize and evaluate exponential functions with the base a

I. Exponential Functions

Transcendental Function A function which is not an algebraic function. In other words, a function which “transcends,” or cannot be expressed in terms of algebra.

Transcendental Functions: Exponential Logarithmic Trigonometric Inverse Trigonometric

Definition of an Exponential Function

The exponential function f with the base a is denoted by f (x) = a x where a > 0, a  1, and x is any real number.

For instance, f (x) = 3 x and g (x) = 0.5 x are exponential functions.

The value of f (x) = 3 x when x = 2 is f (2) = 3 2 = 9 The value of f (x) = 3 x when x = –2 is f (–2) = 3 –2 = 1919

The value of g (x) = 0.5 x when x = 4 is g(4) = =0.0625

Your Turn

The value of g (x) = 0.5 x when x = -4 is g (-4) = =16 Don’t use a calculator!

Example 1. Evaluating Exponential Expressions

Using a calculator, evaluate the expression to 1/ [(1/2) -3.2 ]= x  y / - = *  (-)3.2 * 156 OR

Your Turn

Using a calculator, evaluate each expression to 1/ ( ) =  = = ,

GOAL II. To graph exponential functions

II. Graphs of Exponential Functions

Standard Students understand exponential functions (graphs).

Example 2. Graphs of y = a x

xf(x)f(x)(x, f(x)) Example: Graph f(x) = 2 x Sketch the graph of f (x) = 3 x. 1/91/9 (-2, 1 / 9 ) /31/3 (-1, 1 / 3 ) (0, 1) (1, 3) (2, 9)

Sketch the graph of f (x) = 3 x. x xf(x)f(x) -2 1/91/9 1/31/ –2 2 4 Example: Graph f(x) = 2 x 8 y

Your Turn

xf(x)f(x)(x, f(x)) Example: Graph f(x) = 2 x On same graph, sketch f (x) = 4 x. 1 / 16 (-2, 1 / 16 ) /41/4 (-1, 1 / 4 ) (0, 1) (1, 4) (2, 16)

Add the graph of f (x) = 4 x. x xf(x)f(x) -2 1 / 16 1/41/ –2 2 4 Example: Graph f(x) = 2 x 8 y

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, a > 1 y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (a > 1) 4 4

Example 3. Graphs of y = a -x

xf(x)f(x)(x, f(x)) Example: Graph f(x) = 2 x Sketch the graph of f (x) = 3 -x. 9(-2, 9) 3 -(-2) 3 -(-1) 3 (-1, 3) (0, 1) 3 -11/31/3 (1, 1 / 3 ) 3 -21/91/9 (2, 1 / 9 )

x 6 2 –2 4 Sketch the graph of f (x) = 3 -x. xf(x)f(x) /31/3 2 1/91/9 8 y

Your Turn

x 6 2 –2 4 Which is the graph of f (x) = 4 -x ? 8 y a b a or b

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, 0 < a < 1 y x ( 0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (0 < a < 1) 4 4

Transformations of y = b  a x-c or y =  a x-c + b

The graph of f (x) = 4 x. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

negative If a is negative the graph is reflected over the x-axis.

The graph of f (x) = 4 x and g (x) = -4 x x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

What is the domain and range of f (x)? D = (- ,  ) R = (0,  )

What is the domain and range of g (x)? D = (- ,  ) R = (- , 0)

negative If x is negative the graph is reflected over the y-axis.

The graph of f (x) = 4 x and g (x) = 4 -x x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

The graph of f (x) = 4 x-0. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

When you subtract a positive number c from x, you are translating horizontally the graph of the function c units to the right. If c = 1, then x – 1, so move 1 unit right

The graph of f (x) = 4 x-0 and g (x) = 4 x-1 g (x) = 4 x-1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

When c is negative, you are translating horizontally the graph of the function |c| units to the left. If c = -1, then x - (-1) = x + 1, so move 1 unit left

The graph of f (x) = 4 x-0 and g (x) = 4 x+1 g (x) = 4 x+1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

When you add a positive number b to a function, you are translating vertically the graph of the function b units upwards. If b = 1, move up 1 unit.

The graph of f (x) = 4 x and g (x) = 4 x + 1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

When b is negative, you are translating vertically the graph of the function |b| units downwards. If b = -1, move down 1 unit.

The graph of f (x) = 4 x and g (x) = 4 x - 1. x 2 2 –2 -2 Example: Graph f(x) = 2 x 4 -4 y

Example 4. Transformations of Graphs of Exponential Functions

Describe the movement of the transformation of f (x) = 5 x x x – x + 2

1.-5 x x – x left and reflected over x-axis 3 down 2 up and reflected over y-axis

Your Turn

Match the exponential function with its graph. 1.6 x x x

a 2

b 3

c 1

What has six wheels and flies?

A garbage truck! What has six wheels and flies?

GOAL III. To recognize and evaluate exponential functions with base e

III. The Natural Base e

e  The natural exponential function is y = e x

Example 5. Evaluating the Natural Exponential Function

Use a calculator to evaluate each expression to the nearest 1/ e e

Your Turn

Use a calculator to evaluate each expression to the nearest 1/ e e e e e 0

Example 6. Graphing Natural Exponential Functions

Use the data above, graph the function y = 2e 0.15x

xy=2e 0.15x y -2 2e e e e e

Your Turn

Graph the function y = ½ e -0.6x

x y= ½ e -0.6x y -2 ½ e ½ e ½ e ½ e ½ e

GOAL IV. To graph exponential functions to model and solve real-life applications

IV. Applications

Compounding n times per year Interest Formula A = P 1 + rnrnnt

A = amount earned P = principal = amount originally invested r = rate as a decimal n = # of compoundings per year t = number of years

Compounding continuously Interest Formula A = Pe rt

A = amount earned P = principal = amount originally invested r = rate as a decimal t = number of years

Example 7. Compound Interest

Complete the table to determine the balance (A) for $1000 invested at 6% for 10 years.

n P(1+(r/n)) nt A (1.06) (1.03) (1.015) (1.005) (1.----)

How about continuously? A = 1000e (.06●10) A = Pe rt = $

Your Turn

Complete the table to determine the balance (A) for $1000 invested at 3% for 10 years.

n P(1+(r/n)) nt A Con 1000(1.03) (1.0025) (1.----) e

Example 8. Radioactive Decay

Let Q (in grams) represent a mass of carbon-14 ( 14 C), whose half-life is 5730 years. The quantity of carbon-14 present after t years is Q = 10 t/

Determine the initial quantity (when t = 0). Q = /5730 = = 10(1) = 10 g

Determine the quantity present after 2000 years. Q = /5730 = 7.85 g

Your Turn

Exponential functions can be used to model the amount of defoliation caused by the gypsy moth.

To estimate the amount of defoliation caused by the gypsy moth during a given year, a forester counts the number x of egg masses on 1 / 40 of an acre (circle of radius 18.6 feet) in the fall.

To find the percent of defoliation y the next spring, he uses the equation y = which he derived from the actual data e x

Estimate (to the nearest percent) the defoliation if 36 egg masses are counted on the target acre. y = e (36) = 64.7%

What is a trees least favorite month?

Sep–timber!