7.1 Exploring Exponential Models
Exponential function: y = abx, a ≠ 0 and b < 0 and b ≠ 1.
Exponential growth: a > 0 and b > 1
Exponential decay: a > 0 and 0 < b < 1
Asymptote: a line that a graph approaches as x or y increases
in absolute value (does not touch)
Growth factor: b when b > 1
Decay factor: b when 0 < b < 1
y-intercept: (0, a)
asymptote: y=0
domain: all real numbers
range: y > 0
To: Graph:
1) Make a table
2) Plot and connect points
3) Should look like red or blue line:
Examples:
Graph each function. *leave room for graph*
1) y= 3x
2) r(p)= 3(2)p
Without graphing, answer: decay or growth? y-intercept? (1 line)
3) y = 12(0.95)x
4) y = 0.25(2)x
5) y = 3(4x)
You put $1000 in a college savings account. The account pays 5% interest annually.
6) How much money after 3 years?
Write an exponential function to model the situation. Find the amount after the specified time.
7) A population of 58,000 grows 3.5% per year for 10 years
7.2 Properties of Exponential Functions
Natural base exponential functions: “e” as base e≈2.71… *use calculator to evaluate*
Examples:
Graph each function as a transformation of its parent function. (Leave room for graph)
1) y=2x -3
2) y= 2x+2
Evaluate each expression to 4 decimal places. (Leave 1 line)
3) e-5
4) e3/2
5) e2
Find the amount in a continuously compounded account for the given conditions. (2 lines)
6)
principal:$3000
annual interest rate: 5%
time: 4 years
7)
principal: $1500
annual interest rate: 4.5%
time: 15 years
7.3 Logarithmic Functions as Inverses
“BAE”: Base, Answer, Exponent
Example: Logarithmic Form
Exponential Form
log28 = 3
23=8
Common: (base 10) log10x = logx
Natural: (base e) logex = lnx
*inverses of each other*
Steps to Evaluate:
-Put in exponential form
-Match bases using common powers
-When bases are same, exponents are equal.
-Set equal to each other and find value of variable
Examples:
(1 line)
(1 line)
(4 lines)
Find the inverse of each function.
1) y= log4x
2) y= log10x
3) y= log (x + 1)
7.4/7.6 Properties of Logarithms
Natural Log Rules:
ln(mn) = ln(m) + ln(n)
or
ln(m/n) = ln(m) – ln(n)
ln(mn) = n ∙ ln(m)
SINGLE LOGARITHM (left side) = EXPANDED FORM (right side)
*MUST BE SAME BASE to use log properties/rules*
Change of base formula:
**base on bottom (BB) **
(Then you can evaluate on calculator since it is “log base 10” or just “log”)
Examples:
Write each expression as single logarithm (1 line each).
1)
2)
3)
4)
5)
log42 + log48
log624 – log64
5 log 3 + log 4
4 ln 3
ln 18 – ln 10
Expand each logarithm (1-2 lines each).
6) log749xyz
7) log3(2x)2
𝑎 2 𝑏3
)
𝑐4
8) log(
Use the Change of Base Formula to evaluate each expression (2 lines each).
9) log29
10) log510
Use the properties to evaluate each expression. (2-3 lines each)
11) log216– log24
12) log 5 + log 40
7.5/7.6 Notes: Exponential and Logarithmic Equations
If b≠1, then by=bx if x=y
*Get bases to match
*Make exponents equal each other
*Solve
…If you can’t get bases to match easily:
*Get left side to only have an exponent
*Take log of both sides
*match base to log base because logbb=1 and If b≠1, then logby=logbx if x=y
*(if e use ln)
*Use change of base formula to solve
*Solve for variable
…If you start with log:
*Isolate log and base (add/subtract, multiply/divide)
[May need to use log properties to make single logarithm]
*Put in exponential form (BAE still applies)
*With ln & e, you may need to use scientific notation if number is very big/small.
*If you started with multiple variables, you must check answer for extraneous solutions!
** (you can’t take log of a negative) ** so, if it’s negative when plugged in, that answer is extraneous
Examples: Solve Each Equation
2 lines:
1) 2x=8
2) 25x+1=32
3) 23x=4x+1
4 lines:
4) 2x=3
5) 8+10x=1008
6) 252x+1=144
7) ln x = -2
8) ln4r2=3
9) log x + 4 = 8
10)
3 log x = 1.5
11)
log (5–2x) = 0
12)
𝑒x=18
𝑥
13)
14)
15)
7–2𝑒 2 = 1
log 5 – log 2x = 1
log (x – 3) + log x = 1