Section 3.2: The Logarithm
Recall the way inverse functions have the property of ‘undoing’ one another.
Plug in x = 5
output 125, plug into inverse
f ( x) = x 3
Plug in x = 10
inverts back to 5
f −1 ( x ) = 3 x
output 40, plug into inverse
inverts back to 10
f −1 ( x ) = x / 4
f (x) = 4 x
The inverse operations you’ve experienced up to this point:
Inverting addition with subtraction
Inverting multiplication with division
Inverting powers with roots
... and now it’s time to learn how to invert an exponential expression.
Plug in x = 4
output 81, plug into inverse
inverts back to 4
f −1 (x) = log 3 (x)
f (x) = 3x
First and foremost ...
LOGARITHMS WERE CREATED TO
INVERT EXPONENTIAL FUNCTIONS!!
Exponential expression
x = ay
can be written in an
equivalent inverse form
Logarithmic expression
y = log a ( x)
Knowing how to rewrite between the two forms will really help you understand
what logarithms are; LOGARITHMS ARE EXPONENTS!
ex) Evaluate these logarithms by rewriting them in an equivalent exponential form
When asked to evaluate a logarithmic expression like log a ( X ) ask yourself;
“What exponent or power to I need to raise a to in order to get X ?”
Logarithmic Form
Exponential Form
Value of “?”
log 4 (64) = ?
Pronounced:
“log base 4 of 64”
log 5 (5) = ?
log7 ( 491 ) = ?
log 3 (1) = ?
log(100,000) = ?
This is the common logarithm and
its base is understood to be 10.
It’s the _LOG_ button on your calculator.
The Natural Logarithm is the logarithm that uses base e.
ln(x) = log e (x) It’s the _LN_ button on your calculator.
ex) Evaluate log(21750) on the calculator and round to 5 decimal places.
What does this number represent?
log(21750) ≈ ________________
Evaluate ln(496) on the calculator and round to 5 decimal places.
What does this number represent?
ln(496) ≈ ________________
The Graph of the Logarithmic Function
REMEMBER: The logarithmic function is designed to be the exponential inverse.
Exponential Function
y = 2x
... and its ...
Logarithmic Inverse
y = log2 (x)
REMEMBER HOW
THE COORDINATES
CAN BE REVERSED
TO MAKE THE
INVERSE GRAPH
Domain: __________
Domain: __________
Range: __________
Range: __________
y‐intercept @ __________
x‐intercept @ __________
Horizontal
Asymptote @ __________
Vertical
Asymptote @ __________
ex) Sketch the graph of y = log 3 ( x) and provide its graph summary information.
Domain: __________
Range: __________
Intercepts: x __________
y __________
Asymptote equation: ________
ex) Use transformations to sketch the following graphs.
y =−log 3 (x + 2)
y = log 3 ( x) + 2
Domain: __________
Range: __________
Intercepts: x __________
y __________
Asymptote equation: ________
Domain: __________
Range: __________
Intercepts: x __________
y __________
Asymptote equation: ________
Cancelation Properties of Logarithms
REMEMBER inverse functions cancel each other when composed: f ( f −1 (x)) = x
ex: Plugging a cube root into a cubing function: ( 3 x )3 = x
Plugging a doubling function into a halving function: (22x ) = x
If you plug exponential base a into logarithmic base a you get:
log a (a x ) = x also true for e and natural log ln(e x ) = x
If you plug logarithmic base a into exponential base a you get:
aloga ( x ) = x also true for e and natural log eln( x ) = x
ex) Evaluate the following using the cancelation properties
log 8 (8−2 / 3 )
log(0.0001)
ln( e2000 π )
eln(1234)
62log6 (70)