Objectives
Logarithmic Functions
The Basics
Flashback
Consider the graph of the exponential function
y = f(x) = 3x.
• Is f(x) one-to-one?
• Does f(x) have an
inverse that is a
function?
• Find the inverse.
– Change from logarithmic to exponential form.
– Change from exponential to logarithmic form.
– Evaluate logarithms.
– Use basic logarithmic properties.
– Graph logarithmic functions.
– Find the domain of a logarithmic function.
– Use common logarithms.
– Use natural logarithms.
Inverse of y = 3x
f (x) = 3x
y = 3x
x = 3y
Now, solve for y.
y= the power to which 3 must be raised in
order to obtain x.
Symbolically, y = log 3 x
“The logarithm, base 3, of x.”
Logarithm
For all positive numbers b, where b ≠ 1,
Logbx is an exponent to which the base b
must be raised to obtain x.
Domain Restrictions
for Logarithmic Functions
• Since a positive number raised to an exponent
(positive or negative) always results in a positive
value, you can ONLY take the logarithm of a
POSITIVE NUMBER.
• Remember, the question is: What POWER can I
raise the base to, to get this value?
• DOMAIN RESTRICTION:
1
Find the domain of function.
Common Logarithms
Logarithms, base 10, are called common
logarithms.
Write on the top of your test!!
Example
• Find each of the following common logarithms on a
calculator.
Round to four decimal places.
a) log 723,456
b) log 0.0000245
c) log (−4)
Log button on your calculator
is the common log.
Natural logarithms
• Logarithms, base e, are called natural logarithms.
• ln(x) represents the natural log of x, which has a
base=e
• What is e? If you plug large values into
you get closer and closer to e.
• logarithmic functions that involve base e are found
throughout nature
• Calculators have a button “ln” which represents the
natural log.
Example
• Find each of the following natural logarithms on a
calculator.
Round to four decimal places.
a) ln 723,456
b) ln 0.0000245
c) ln (−4)
2
Example
Logarithmic Conversions
• Convert each of the following to a
logarithmic equation.
a) 25 = 5x
b) ew = 30
Rewrite the following exponential
expression as a logarithmic one.
Summary of Properties of Logarithms
• Convert each of the following to an
exponential equation.
a) log7 343 = 3
b) logb R = 12
Finding Logarithms
• Find each of the following logarithms.
a) log2 16
b) log16 4
c) log10 0.001
Example
3