How do I solve x = 3y ?
John Napier was a Scottish theologian and
mathematician who lived between 1550 and
1617. He spent his entire life seeking
knowledge, and working to devise better ways
of doing everything from growing crops to
performing mathematical calculations. He
invented a new procedure for making
calculations with exponents easier by using
what he called logarithms. A logarithm can be
written as a function y = logbx.
Introduction to the
Logarithmic Function
The notation y = logbx is another way of writing x = by.
So x = by and y = logbx represent the same functions.
• y =log3x is simply another way of writing x = 3y.
The notation is read
Logarithmic Function
(Common)
f ( x) = log10 x
Logarithmic Function
(Natural)
f ( x) = ln x
Calculator: y1= log(x)
Calculator: y1= ln(x)
Domain: x > 0
Domain: x > 0
Range: y ε 
Range: y ε 
Zeros: (1,0) or x = 1
Zeros: (1,0) or x = 1
X-Intercept: (1,0)
X-Intercept: (1,0)
Vertical Asymptotes: x = 0
Vertical Asymptotes: x = 0
Logarithmic Function
“y is equal to the logarithm, base 3, of x.”
Logarithmic Function
Reading a logarithmic function
log b y = x
Logarithmic Form
expression log b y = x is read as “log base b
of y equals x”.
The expression log 6 36 = 2 is read as “log base 6
of 36 equals 2”.
The common log function will always have a
base of 10, this is an understood value. If
there is no “b” value present, you should
assume the function has a base of 10.
bx = y
Exponent Form
The
Logarithmic Form
(exp) = log(base)(product)
EX 1:
y = log525
Exponential Form
(base)(exp) = (product)
5y = 25
Logarithms are Exponents – you can convert
logarithmic functions to exponential functions
and vice versa because the inverse of a log
function is an exponential function.
1
Natural Logarithmic Function
ln e y = x
e =y
Natural Logarithmic Form
What happens with these?
x
Natural Logarithmic functions are the same as
logarithmic functions, but the only difference is
the base is always
e.
2.
log10, 000 = 4
3.
log 2
4.
log 5 1 = 0
1
2.
ln e =
1
3.
log 4 4 =
1
4.
log 37 37 =
1
Write the equation in the equivalent logarithmic form.
Write the equation in the equivalent exponential form.
log 7 49 = 2
log10 10 =
Examples
Examples
1.
1.
Exponent Form
1
= −3
8
log8 64 = 2
1.
82 = 64
2.
5−2 =
1
8
3.
16 2 = 4
log16 4 =
50 = 1
4.
e4 = 54.598
ln 54.598 = 4
7 2 = 49
104 = 10, 000
2 −3 =
log 5
1
= −2
25
1
1
2
Logarithms are Exponents
Ex 2 Logarithms are Exponents
Complete the table below for the function x = 2y.
The inverse of this equation is y = 2x .
1
25
Since these graphs are inverses we should also know that the
y=x
graphs will be reflections of each other over the line ________.
f(x)
When you convert the exponential equation to a logarithm
equation (done by finding the inverse) you get log2x = y.
Enter both of those equations in the calculator. The tables
should be reversed since the functions are inverses of each
other.
x
Table 1
X
F(x)
1
2
2
4
3
8
Table 2
X
F(x)
2
1
4
2
8
3
What is the relationship between the two graphs?
Inverses
2
CONT…
Examples
The inverse of an exponential function is
a __________
Logarithmic function?
The inverse of y = 2 x is the function
y = log2 x (read as Log base 2 of x). If the
function had been y = 3 x, its inverse
would have been
y = log 3 x
_________________?
Explain the difference between
Equivalent Functions
Write the equation in the inverse form.
1.
y = 4x
y = log 4 x
2.
y = 6x
y = log 6 x
3.
y = log 2 x
y = 2x
4.
y = log 7 x
y = 7x
Equivalent Equations
x = 3y and y = log3x
Inverse Functions
Only One Graph Is Visible.
Inverse Equations
The inverse of the exponential parent function can be defined as a new
function, the logarithmic parent function. The functions are reflections
of each other over the line y = x.
y = 2x and y = log2x
Two graphs are visible that are reflected over the y = x line.
3