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HONORS ALGEBRA 2 EXPONENTIALS & LOGARITHMIC FUNCTIONS
Lessons 6.1-6.2: Exponential Growth/Decay
Exponential functions are of the form: y = abx
If 0 < b < 1, the graph represents exponential DECAY.
If b > 1, the graph represents exponential GROWTH.
Graph: y = 2x + 3 (table of values)
The Number e
e is an irrational number approximately
equal to 2.71828
Exponential functions with a base of e
are useful for describing continuous
growth or decay.
Growth factor: b = 1 + r
Decay factor: b = 1 – r
1
2
𝑥
Half-life: y = a( )ℎ𝑎𝑙𝑓−𝑙𝑖𝑓𝑒
Compound Interest formula:
r
A  P(1  ) nt
n
Continuously Compounded Interest formula
A = balance after t years
P = principal
r = rate
n = # times interest is added per year
t = time in years
A = Pert
A = amount in account,
P = principal,
r = annual rate of interest,
t = time in years
Ex 2: Suppose you invest $1,050 at an annual interest rate
of 5.5% compounded continuously. How much will you have
in the account after five years?
Ex 1:How much must you deposit in an account that pays 8%
annual interest, compounded monthly, to have a balance of
$1,500 after one year?
Lesson 6.3: Logarithms
A LOGARITHM IS AN EXPONENT.
Write the equation in exponential form:
log464 = 3 ___________________
logbx = y and by = x are
equivalent expressions
Write the equation in logarithmic form:
24 = 16 _____________________
Evaluate the logarithms:
1)
log 16
2)
4)
5)
2
log39
log497
log225
3)
6)
log88
log31
A natural logarithm (ln) is a log with base e. All the same properties of logs also apply to natural
logs.
Simplify using mental math:
1) ln e
2) ln e3
3) ln 1
4) 3 lnx + ln y
Lesson 6.5: Properties of Logarithms
PROPERTIES OF LOGARITHMS
Let a, u, and v be positive numbers such that a ≠ 1, and let n be any real number.
Product Property:
Quotient Property:
Power Property:
loga uv 
u
log a 
v
loga u n 
Note: There is no property to simplify the logarithm of a sum.
USE PROPERTIES OF LOGS TO EXPAND AND CONDENSE
Use the properties of logarithms to simplify (condense): (Write as one log expression)
1) 5log3 + 2log2
Use the properties of logarithms to expand:
(Write as a sum or difference of logarithms, or use power property)
2) log45 𝑥
Lesson 6.6: Exponential & Logarithmic Equations
Change of Base formula:
log a x 
log10 x
log10 a
Steps to solve Exponential Equations:
Steps to solve Logarithmic Equations:
1) Isolate power
2) Take log of both sides in same base
3) logaa = x
1) Use properties of logs, if necessary to
condense log expression.
4) Use change of base formula
a. Solve: 2 + 3x = 4
b. log (3x + 1) = 5
2) Write in exponential form
c. ln 3 + lnx = 6
d. e 2x + 1 = 6