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Level 409

Exponential & Logarithmic Functions II


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Asymptote
An imaginary line on a graph that acts as a boundary line.
Base
the number that is written with an exponent
Common logarithm
A logarithm whose base is 10, or just log
Exponential equation
An equation that contains one or more exponential expressions
Exponential Function
y = ab^x
Exponential regression
A statistical method used to fit an exponential model to a given data set
inverse function
Let f be a one-to-one function with domain A and range B. Then its ________________ f^-1 has domain B and range A and is defined by f^-1 (y) = x, f(x)=y for any y in B
Logarithm
The exponent that a specified base must be raised to in order to get a certain value
Logarithmic equation
An equation that contains a logarithm of a variable
Logarithmic regression
A statistical method used to fit a logarithmic model to a given data set
Natural logarithm
The logarithm with base e is called the natural logarithm and is denoted by "ln"
Natural logarithmic function
The function f(x)=ln x
change to exponential form
How to solve a log equation
Logb 1
= 0
X^0
= 1
Logs with "no base"
*log with no base is always base 10
When do you simplify a log?
*fully solve the log whenever you can
No solution log functions
*a positive base with a negative log always= no solution
*always put parenthesis around what you write/ do first
How to write a long when you have both (÷) and (X) or (+) and (—)
*power/root
When/ how to convert from a radical in a log to a fractional exponent
*when log is expanded, exponent goes in front of log
Where is an exponent written in expanded log form, and where it is written in a single log?
Converting from log form to exponential
*in log equations, you are trying to find the exponent
y= #logb X
Graphing logs: compression/ stretch
Graphing logs: horizontal shift
y= logb (X - #)
Graphing logs: vertical shift
y= logb X + # OR y= logb (X-#) + #
parent graph of log functions:
What is the domain/ range of parent graph of log function, and how do you find domain/range if they change?
*only changes when there's a horizontal shift
When does the Domain of a log function change?
*only changes when there's a vertical shift
When does the Range of a log function change?
For exponentials
Exponential Growth vs. Exponential Decay
Domain and Range of Exponentials
*unless there is a transformation*
parent graph:
Transformations with exponential functions
y intercept of an exponential for parent graph
*unless there is a transformation y intercept= (0,1)
y= ab^-x
*not the same as negative in front of whole function (reflection across x axis)
How to get the value of x in exponentials using the graph
•if the function= #, find what value x is when that # is y
a=c
If b^a= b^c
Change of base
logb X= log (x)/ log (b)
e
*continuous growth factor≈ 2.718
Base e
*base e= ln
Solve exponentials w/e
*to solve exponentials with e, take the ln of both side
y= lnx
Inverse of y= e^x
Compounded Continuously Formula
A= P • e^rt
Compounded Annually Formula
y= a(1+r) ^t
10^m1-m2
Earth Quake Intensity
if g(x) is the inverse of the function f(x)
the f(g(x)) is equal to x and g(f(x)) is equal to x
f^-1(x)
to denote the inverse function of f(x) write
to determine if a function has an inverse or is invertable
use a horizontal line test. For a function to have an inverse any y value must only map to one x-value.
one to one function
a function that has an inverse that is also a function
Horizontal Line Test
inverse of a function is also a function if and only if no horizontal line intersects more than one
Graphically two functions are inverse of each other if
they are a reflection of each other over the y = x line
Algebraically two functions are inverses of each other if
f o f-1(x) = x and f-1 o f(x) = x
f composed of f inverse
f o f-1(x) is read as
first identify several points of the graph f,
To reflect the graph of a function f across the y = x line,
To find the inverse of a function
swap the x and y values and solve for y, if the new equation can be solved for y as a function of x, it is the inverse of the original function. (even functions are not invertable)