Exponential and Logarithmic Functions

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Exponential and Logarithmic Functions

• Learning Objective
• What is an exponential function?
• ex
• Understand what a natural logarithm is.
• Transforming
• Inverses

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y 4x
5
y 4x
4
3
2
1
0
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
The x-axis is an asymptote
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y ax
y ax
0
Functions of the form f(x) ax are known are
exponential functions
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y ex
When x0 y e0 1
So it goes through (0,1)
The x-axis is an asymptote
y ex can, of course, be transformed
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y ex and y e-x and y 2ex
y ex
y e-x is a reflection in the y-axis
y 2ex scales by factor 2 in the y-direction
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Inverse Functions
What is the inverse of ?

• Write function as a rule in terms of y and x.
• Swap x and y
• Rearrange to get in terms of y.

y 3x
x 3y

• log x log 3y
• log x y log 3
• y log x / log 3

INVERSES OF EXPONENTIALS ARE LOGS!!!
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Inverse Functions
What is the inverse of ?

• Write function as a rule in terms of y and x.
• Swap x and y
• Rearrange to get in terms of y.

y ex
x ey

• ln x ln ey
• ln x y ln e
• y ln x

THE INVERSE OF ex is ln
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Natural Logarithms

• Any log to the base e is known as a
• natural logarithm.
• In French this is a
• logarithme naturel
• Which is where ln comes from.
• When you see ln (instead of log)
• then its a natural log

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y ex
y ex
When x0 y e0 1
So it goes through (0,1)
The x-axis is an asymptote
y ex can, of course, be transformed
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y ex and y x
y ex
y x
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y ex, y x and y ln x
FACT inverses of any function are a reflection
in the line yx
y ex
y x
y ln (x)
yln(x) is a reflection of y ex in the line y
x
yln(x) and y ex are inverse functions
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Inverse functions with ex
e.g. f(x) ex -2
y ex -2
y 2 ex
ln(y 2) ln ex
ln(y 2) x
The inverse of f(x) is
f-1(x) ln(x 2)
Domain is x gt -2
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Inverse functions with ln x
e.g. f(x) ln(2x) 6
y ln(2x) 6
y - 6 ln (2x)
ey-6 eln 2x
The inverse of f(x) is
ey-6 2x
f-1(x) ½ ex-6
Domain ?
½ ey-6 x
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y ex and y ln (x1) and y ln 3x
y ln 3x scales by factor 1/3 in the x-direction
y ln (x1) is a translation by -1 in the
x-direction
y ln (x)