GRAPHING EXPONENTIAL FUNCTIONS

f(x) = 2x

2 > 1

exponential growth2

2 4–2

4

6

–4

y

xNotice the asymptote: y = 0

Domain: All real, Range: y > 0

GRAPHING EXPONENTIAL DECAY

x

21)x(f

0 < < 1

exponential decay

12 2

2 4–2

4

6

–4

y

xNotice the asymptote: y = 0

Domain: All real, Range: y > 0

Graph Natural Base Exponential FunctionsUse the graph of f (x) = ex to describe the transformation that results in h (x) = e–x – 1. Then sketch the graph of the function.

Answer: h (x) is the graph of f (x) reflected in the y-axis and translated 1 unit down with a vertical asymptote at x = -1.

Domain: All real, Range: y > -1

Graph f(x) = ex–2 + 1.

Graphing Exponential Functions

VA: x = 1

Domain: All real, Range: y > 1

COMPOUND INTEREST FORMULA

A: amount of the investment at time tP: principalr: annual interest rate as a decimaln: number of times interest is

compounded per yeart: time in years

A(t) = P 1 +

( )rn

nt

FIND THE FINAL AMOUNT OF $100 INVESTED AFTER 10 YEARS AT 5% INTEREST COMPOUNDED ANNUALLY, QUARTERLY AND DAILY.

ANS: $162.89 annually, $164.36 quarterly, $164.87 daily

Recall the compound interest formula A = P(1 + )nt, where A is the amount, P is the principal, r is the annual interest, n is the number of times the interest is compounded per year and t is the time in years.

nr

The formula for continuously compounded interest is A = Pert, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years.

What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously?

Economics Application

The total amount is $4083.08.

A = Pert

Substitute 500 for P, 0.0525 for r, and 40 for t.

A = 500e0.0525(40)

Use the ex key on a calculator.

A ≈ 4083.08

You can write an exponential equation as a logarithmic equation and vice versa.

Logarithmic Form

Exponential Equation

log99 = 1

log2512 = 9

log82 =

log4 = –2

logb1 = 0

116

13

91 = 9

29 = 512138 = 2

1164–2 =

b0 = 1

Evaluate by using mental math.Evaluating Logarithms by Using Mental Math

The log is the exponent.

Think: What power of 5 is 125?

log5 125

5? = 125

53 = 125

log5125 = 3

Exponential and logarithmic operations undo each other since they are inverse operations.

Simplify.

a. ln e3.2 b. e2lnx

c. ln ex +4y

ln e3.2 = 3.2 e2lnx = x2

ln ex + 4y = x + 4y

Graphs of Logarithmic Functions

Sketch and analyze the graph of f (x) = log2 x. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

Graphs of Logarithmic Functions

Answer: Domain: (0, ∞); Range: (–∞, ∞); x-intercept: 1; Asymptote: y-axis; Increasing: (0, ∞); End behavior: ;

Expand Logarithmic Expressions

A. Expand ln 4m3n5.

Answer: ln 4 + 3 ln m + 5 ln n

Expand Logarithmic Expressions

Expand .

Condense Logarithmic Expressions

Condense .

Answer:

Use the Change of Base Formula

Evaluate log 6 4.

log 6 4 = Change of Base Formula≈ 0.77 Use a calculator.

Answer: 0.77

Solve Logarithmic Equations Using One-to-One Property

Solve 2 ln x = 18. Give exact and round to the nearest hundredth.

2 ln x = 18ln x = 9eln x = e9

x = e9

x ≈ 8103.08

Solve 7 – 3 log 10x = 13. Round to the nearest hundredth.

7 – 3 log 10x = 13

–3 log 10x = 6log 10x = –210–2 =10x10–3 = x

= x

Log Circle at this point.

Solve log2 5 = log2 10 – log2 (x – 4).

log25 = log210 – log2(x – 4)log25 =

5 =

5x – 20 = 105x = 30x = 6

Solve Exponential Equations

Solve 3x = 7. Round to the nearest hundredth.

3x = 7log 3x = log 7x log 3 = log 7

x = or about 1.77

When the variable is the exponent, take the log/ln of both sides.

Solve Exponential Equations

Solve e2x + 1 = 8. Give exact and round to the nearest hundredth.

e2x + 1 = 8ln e2x + 1 = ln 8

2x + 1 = ln 8

x = or about 0.54

Solve log (3x – 4) = 1 + log (2x + 3).log (3x – 4) = 1 + log (2x + 3)

Check for Extraneous Solutions

log (3x – 4) – log (2x + 3)= 1

= 1

Check for Extraneous Solutions

= 10

3x – 4 = 10(2x + 3)3x – 4 = 20x + 30–17x = 34x = –2Since neither log (–10) or log (–1) is defined,

x = –2 is an extraneous solution.

Answer: no solution

Solve and check.4x – 1 = 5

log 4x – 1 = log 5

5 is not a power of 4, so take the log of both sides.

(x – 1)log 4 = log 5

Apply the Power Property of Logarithms.

Solving Exponential Equations

Divide both sides by log 4.

x = 1 + ≈ 2.161

log5log4

x –1 = log5log4

Exact and approximate answers

Solve.

Solving Logarithmic Equations

Write as a quotient.

log4100 – log4(x + 1) =

1

x = 24

Use 4 as the base for both sides.

Use inverse properties on the left side.

100 x + 1log

4( ) = 1

4log4 = 41

100x + 1( )

= 4 100 x + 1

DOUBLING YOUR INVESTMENT.How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously?

How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?

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