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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