Logarithms
A logarithm function is the inverse function of an exponential function.
Exponential Function
Logarithmic Function
y = bx
x = by
(exponential form)
¾ Domain: All real numbers
¾ Range: y > 0
¾ Graph has a horizontal
asymptote.
↔ log b x = y
(logarithm form)
¾ Domain: x > 0
¾ Range: All real numbers
¾ Graph has a vertical
asymptote.
Examples: Rewrite each logarithm into its exponential form.
1
1. log 3 27 = 3
2. log 41024 = 5
3. log 2 = -3
8
Examples: Write each log in exponential form to find each logarithm.
1. log464
2. log2128
3. log322
Log of 1:
¾ log b 1 = 0 Æ b0 = 1
¾ EX) log 41= 0
Special Logarithms
Log of b with base b:
¾ log b b = 1 Æ b1 = b
¾ EX) log 6 6 =1
Common Log: Log with base 10
¾ The is the LOG key on your
calculator
¾ Log 5 ≈ ________
Natural Log: Log with base e
¾ This is the ln (LN) key on
your calculator.
¾ EX) ln 10 ≈ ________
Inverses of Log Functions
Reminder: Exponential Functions and Log Functions are inverses of
each other.
9 Switch x and y.
9 Solve for y by rewriting in exponential form.
Example: Find the inverse of each log function.
2. y = ln x +5
1. y = log3 (x – 2)
Graphing Log Functions
Graphs of log functions are inverses of exponential growth/decay
functions.
1. Graph the parent graph: y = log b x
™ Make an x-y table for the exponential function y = bx , like we did
in Section 8.1 and 8.2. This is the inverse of the log function.
™ Since the exponential function is the inverse of the log function,
switch the x and y values in your x-y table and graph them.
™ Draw a smooth curve using your points remembering to not cross
the y-axis which is now the vertical asymptote.
2. Translate the graph according to h (left or right) and k (up or down).
Note: If you translate left or right, make sure to draw the new vertical
asymptote.
Example: Graph each function. Then, state the domain and range.
1. y = log3 (x – 2)
2. y = log1/2 x + 4
y
y
10
10
5
5
x
-5
-10
-10
x
-5
-5
5
10
-10
-10
-5
5
10