Lesson 5-5
Logarithms
Logarithmic functions
Logarithmic functions
The inverse of
the exponential function.
Logarithmic functions
The inverse of
the exponential function.
Basic exponential function:
f(x) = bx
Logarithmic functions
The inverse of
the exponential function.
Basic exponential function:
f(x) = bx
Logarithmic functions
The inverse of
the exponential function.
Basic logarithmic function:
f-1(x) = logbx
Logarithmic functions
The inverse of
the exponential function.
Basic logarithmic function:
f-1(x) = logbx
Logarithmic functions
The inverse of
the exponential function.
Basic logarithmic function:
f-1(x) = logbx
Every
(x,y)  (y,x)
Logarithmic functions
Basic rule for changing exponential
equations to logarithmic equations
(or vice-versa):
Logarithmic functions
Basic rule for changing exponential
equations to logarithmic equations
(or vice-versa):
logbx = a  ba = x
Logarithmic functions
Basic rule for changing exponential
equations to logarithmic equations
(or vice-versa):
logbx = a  ba = x
The base of the logarithmic
form becomes the base of the
exponential form.
Logarithmic functions
Basic rule for changing exponential
equations to logarithmic equations
(or vice-versa):
logbx = a  ba = x
The answer to the log
statement becomes the power
in the exponential form.
Logarithmic functions
Basic rule for changing exponential
equations to logarithmic equations
(or vice-versa):
logbx = a  ba = x
The number you are to take
the log of in the log
form, becomes the answer in
the exponential form.
Examples:
Examples:
log525 = 2 because 52 = 25
Examples:
log525 = 2 because 52 = 25
log5125 = 3 because 53 = 125
Examples:
log525 = 2 because 52 = 25
log5125 = 3 because 53 = 125
log2(1/8) = - 3 because 2-3 = 1/8
base b exponential function
f(x) = bx
base b exponential function
f(x) = bx
Domain: All reals
Range: All positive reals
base b logarithmic function
f-1(x) = logb(x)
base b logarithmic function
f-1(x) = logb(x)
Domain: All positive reals
Range: All reals
Types of Logarithms
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
log10(x)  called the common
logarithm
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
log10(x)  called the common
logarithm
For the common logarithm
we do not include the
subscript 10, so all you will
see is: log (x)
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
So, log10(x)  log (x) = k if
10k = x
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
loge(x)  called the natural
logarithm
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
loge(x)  called the natural
logarithm
For the natural logarithm, we
do not include the subscript
e, so all you will see is:
ln (x)
Types of Logarithms
There are two special logarithms
that your calculator is programmed
for:
So, loge(x)  ln (x) = k
if ek = x
Examples:
Examples:
log 6.3 = 0.8 because 100.8 = 6.3
Examples:
log 6.3 = 0.8 because 100.8 = 6.3
ln 5 = 1.6 because e1.6 = 5
Example:
Example:
Find the value of x to the nearest hundredth.
Example:
Find the value of x to the nearest hundredth.
Example:
Find the value of x to the nearest hundredth.
10x = 75
Example:
Find the value of x to the nearest hundredth.
10x = 75
This transfers to the log statement
log 10 75 = x
and the calculator will tell you
x = 1.88
Example:
Find the value of x to the nearest hundredth.
ex = 75
Example:
Find the value of x to the nearest hundredth.
ex = 75
This transfers to the log statement
ln 75 = x
and the calculator will tell you
x = 4.32
Evaluate:
Evaluate:
log8 2
Evaluate:
1
ln 3
e
Evaluate:
1
log
10, 000
Evaluate:
log5 1
Solve:
Solve:
log x = 4
Solve:
1
ln x =
2
Solve:
log x = -1.2
Assignment:
Pg. 194
C.E. #1 – 9 all
W.E. #2 – 14 evens