Algebra 2 AII.6 Exponential/Logarithmic Functions Notes
Mrs. Grieser
Name: _____________________________________ Date: _______________ Block: _______
Exponential Functions

Exponential functions are functions made of exponential expressions where the base is a
constant and the exponent is variable.

Which is an exponential function (circle)? f(x) = x2
g(x) = 2x
Exponential functions are of the form
f(x) = ax
where a is a constant and x is a variable

Graph: f(x) = 2x
o Domain?
o Range?
o y-intercept?
o x-intercept?

The graph has an asymptote. Describe what an
asymptote is:
_________________________________________________

Where is the asymptote for this function?
_____________________________________
a) Graph f(x) = 3x
b) Graph f(x) = 4x + 1
How are the graphs similar or different? _________________________________________________

Exponential functions show growth or decay.
o Grow (or decay) much faster than linear functions.
o Examples of real life situation that grow exponentially: ___________________
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Algebra 2 AII.6 Exponential/Logarithmic Functions Notes
Mrs. Grieser
Exponential Transformations

Exponential functions have parent functions, and can be graphed using
transformations.
Graph and compare:
f(x) = 2x
g(x) = 2-x
j(x) = 2x + 2
h(x) = 2x+2
k(x) = -2x
m(x) = -2-x
n(x) = -2-x + 2
Conclusions: for a, a natural number,

y = ax+1
: shifts _________

y = ax - 1 : shifts ________

y = ax - 1 : shifts _________

y = a-x
: reflects _________

y = ax + 1 : shifts _______

y = -ax
: reflects _________
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Algebra 2 AII.6 Exponential/Logarithmic Functions Notes
Mrs. Grieser
Logarithmic Functions

A Logarithm is the exponent of a number

Example: what exponent do you raise 10 to the power of to get 100? __________

Write 102 = 100 as a log ________________________

Write in log form:
a) 23 = 8
b) 42 = 16
c) 34 = 81
Generalization:
logax = y IFF ay = x

Say "log base a of x is y"

If no base is shown, assume base 10:
o log 100 = log10100 = _______

Write in exponential form:
a) log28 = 3
b) log41 = 0
c) log1212 = 1

What can we conclude about logb1 (for any base b)? ___________________________

What can we conclude about logbb (for any base b)? ___________________________
Graphing Logarithmic Functions

Logarithmic functions are inverses of exponential functions
Graph y = 2x

Graph the inverse, x = 2y
x = 2y is the same as y = log2x. Why? _______________________________________________
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Algebra 2 AII.6 Exponential/Logarithmic Functions Notes
Mrs. Grieser
Domain and Range of Logarithmic Functions
y = log2x is shown at right.

Why can x never be ≤ 0?
o Re-write y=log2x in exponential
form_______
o With any natural number base, will any
exponent give you a negative
number?________

Logarithmic functions have a vertical
asymptote at x = 0

Domain______________ Range ____________
Notes on using the graphing calculator

Calculator only can log base 10 or base e (natural log – coming soon!)

For now, use either of these logs since the shape will be similar for any base.
Exponential Transformations
Using y = log2x as a parent function, graph the following functions:
Graph y = log2(x - 2)
Graph y = log2(x + 3)

What are the domain and vertical asymptote for each function?_____________

What happens if you add or subtract from the variable? __________________
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Algebra 2 AII.6 Exponential/Logarithmic Functions Notes
Mrs. Grieser
Using y = log2x as a parent function, graph the following functions:
Graph y = log2(x) - 2

Graph y = log2(x) + 3
What are the domain and vertical asymptote for each function – what changed?
___________________________________________________________________________

What happens if you add or subtract from the function? __________________
Conclusions: for a, a natural number,

y = loga(x + 1) : shifts _________

y = loga(x - 1) : shifts _________

y = loga(x) + 1 : shifts _________

y = loga(x) - 1 : shifts _________
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Algebra 2 AII.6 Exponential/Logarithmic Functions Notes
Mrs. Grieser
You Try…
Graph, identifying domain, range, and asymptotes:
1) y = 4x
2) y = 2x+4 – 3
domain:
domain:
range:
range:
asymptotes:
asymptotes:
3) y = log3x
4) y = log3(x + 1) + 2
domain:
domain:
range:
range:
asymptotes:
asymptotes:
5) Write in logarithmic form:
a) 24 = 16
b) 35 = 243
c) 10y = x
b) log 1000 = 3
c) logxz = y
b) log 1,000,000 = x
c) log6x = 2
6) Write in exponential form:
a) log416 = 2
7) Find x:
a) log525 = x
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