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**Unformatted text preview: **~Log and Exponential Functions~
By Belen Pair
Period 6
Mrs. Taouk ~Graphing~ Graphing: Growth or Decay
b>l = Growth 0<b<1 = Decay *b cannot equal 0 or 1 X- intercept Y- intercept More examples and help: Make y=0 Make X=0 Different Transformations
Reflections
If c is negative, reflect
over the y-axis.
If a is negative, reflect
over the x-axis Shifts
Vertical shift
If k>0, the graph would be
shifted upwards.
If k<0, the graph would be
shifted downwards.
Horizontal Shift
If h>0, the graph would be
shifted left.
If h<0, the graph would be
shifted right. Up, down/left, right
If h is positive go right
If h is negative go left
If y is positive go up
If y is negative go down Stretch/ shrink
If the absolute value of c is
greater than one: horizontal
shrink
If the absolute value of c is
less than one: horizontal
stretch
If the absolute value of a is
greater than one: vertical
stretch
If the absolute value of a is
less than one: vertical shrink ~Graphing Log
functions~
y=a(logbc(x-h))+k
y=a(lnc(x-h))+k
domain → c(x-h)>0
Range → (-∞,+∞)
VA → x=h Inverse Graphs
Context
The graph of inverse function
of any function is the
reflection of the graph of the
function about the line
y=xy=x . So, the graph of the
logarithmic function
y=log3(x)y=log3(x) which is
the inverse of the function
y=3xy=3x is the reflection of
the above graph about the
line y=xy=x . Example:
If k=−3: The graph of y=[log2(x+1)]
will be shifted 3 units down to get
y=[log2(x+1)]−3. Extra Help:
_52xEjT_k
~Graphing
Exponential
functions~ Exponential
Functions
y=a(b^c(x-h))+ k
y=a(e^c(x-h))+ k
Domain→(-∞,+∞)
Range→y>k OR y<k
HA→y=k
Parent Function= b^x or
e^x Y= 2^x Extra help: Example With Harder
Transformation
y= 2^(x+3)
Replacing x with x+3 translates the
graph 3 units to the left.
9DhlR43P7A
iN1cGCgpdQ ~Expand and Condense log
Expressions~ Natural Logs
The natural logarithm of a number is its
logarithm to the base of the e, where e is
an irrational and transcendental number
approximately equal to 2.718281828459.
The natural logarithm of x is generally
written as ln x, loge x, or sometimes, if the
base e is implicit, simply log x.
Parenthesis are sometimes added for
clarity, giving ln(x), loge(x) or log(x).
More Help:
arithmic-functions/natural-logarithms-base-e/nat
ural-logarithms/natural-log-definition Example: ~Expanding~ Log2[(8x^4)/5]
→ When asking you to expand
an equation they want you to
take apart 1 log equation into
multiple. log2(8x^4) - log2(5)
Why Did I Do This?:
I split the numerator and denominator apart
by converting the 1 log with division into Extra help with hard practice problems:
es2.htm two logs using subtraction. Rules: ~Condensing~
Simplifying or combining logs The Product Rule: The logarithm of the product of
numbers is the sum of logarithms of individual numbers.
The Quotient Rule:The logarithm of the quotient of
numbers is the difference of the logarithm of individual
numbers.
The Power Rule: The logarithm of an exponential number
is the exponent times the logarithm of the base.
The Zero Rule: The logarithm of 1 with b > 1 equals zero.
The Identity Rule: The logarithm of a number that is equal
to its base is just 1.
Log of Exponent Rule: The logarithm of an exponential
number where its base is the same as the base of the log
equals the exponent.
Exponent of Log Rule: Raising the logarithm of a number
by its base equals the number Examples For The Most Used Rules
Product Rule
Examples
This is the Product Rule in reverse because
they are the sum of log expressions. We
can convert those addition symbols into
multiplication symbols inside the
parentheses. Quotient Rule
Examples
The difference between logarithmic
expressions tells us to use the Quotient
Rule. I can put together x and 2 inside a
single parenthesis using division
operation. Harder Example:
1. Apply the Power Rule
in reverse to condense
the constants or
numbers on the left of
the logs. 2. Next step is to use the
product and quotient
rules from left to right. Work: Solving Exponential Equations and
Logarithmic Equations One to One
c/log1to1.htm Example:
logbS=logbT if and only if S=T. How to Snail
xponential-and.html Create a snail head around your log base
b. And then you create the snail shell by
raising this b to the power and being set
equal to our last part. Solving Exponents
Some exponential equations
can be solved by rewriting
each side of the equation
using the same base. Other
exponential equations can
only be solved by using
logarithms.
MORE HELP:
xA Example:
solve this
equation -> More Example Practice Problems
1) 3) 2) 4) Solving Logs
Some logarithmic problems
are solved by simply
dropping the logarithms while
others are solved by rewriting
the logarithmic problem in
exponential form.
More Help:
.
htm Example:
solve this
equation -> More Example Practice Problems Solving Exponential and Log Functions
Solving Logs are harder to
teach. Using your
knowledge from one to
one functions and the
snail method, try to
practice solving. Another
Example Quiz: Try it
yourself Word Problems
more help: Example of population growth
Bacteria in a petri dish double in population every hour. Their initial population is 3.
Assuming the absence of limiting factors, how many bacteria are there after 24 hours?
Given:
3 bacteria in a petri dish
They double every hour
24 hours
Want:
How many after 24 hours?
Analysis:
Let A=amount of bacteria after t hours
A=3(2)t
t=24 hours
A=3(2)24
A=50331648 bacteria Example of compound interest
Find the amount of money in an account with holding $200 at 5% interest compounded
semiannually for 8 years.
Given:
Principle amount of money P = $200
Interest rate r = 0.05
Number of times compounded per year n = 2
Time t = 8 years
Amount of money after t years A
Want:
A after 8 years
Know:
A=P (1+r / n)tn
Analysis:
A= 200(1+0.05 / 2)8*2
A=$296.90 Example of Half-Life Problems
Isotope X has a half life of 300 years. After 900 years, how much of an original 200 gram
sample is left?
Given:
300 year half life
900 years
Original 200g
Want:
How much is left after 900 years?
Analysis:
Let t = number of half lives
Let A = amount of isotope after t years
A=200(0.5t)
t=900 years / 300 years = 3 half lives
A=200(0.5)3
A=50 grams left The End ...

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- Winter '13
- Kramer
- Math, Exponential Function, Derivative, Exponential Functions, Natural logarithm, Logarithm