Ch' 3'2: Logarithms and their Graphs What are they

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Ch. 3.2 Logarithms and their GraphsWhat are
they?

• Inverse of the exponential functions
• Definition For x gt 0, a gt 0, and a ? 1, y
  logax if and only if ay x
• y logax is equivalent to ay x, thus a
  logarithm is an exponent
• Ex log28 3 is the same as 23 8
• Ex log3 -4 is the same as 3-4
• Ex Find x if log4x2

x 42 16
x ay, so
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Common log function a 10

• Log key on calculator is log, base 10
• Just enter log then the number when working
  with base 10
• Calculate
• 1. log1054 2. log102.3 3. log10-2

1.7324
0.3617
Not possible
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PROPERTIES

• Logarithm (log)
• loga10 because a0 1
• logaa1 because a1 a
• logaax x because ax ax
• logax logay, then x y
• Natural Log (ln)
• 1. logex ln x
• 2. y ln x if and only if x ey

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Finding the domain of logax

• Take the x-value set it x gt 0 and solve
• REMEMBER THE DOMAIN AFFECTS WHERE THE GRAPH
  WILL OCCUR
• Ex f(x) log4(1-x) Ex g(x) log3x2
• Ex f(x) ln(x 2) Ex g(x) ln x3

x2 gt 0
1 x gt 0
All numbers work Except 0!!
x lt 1
x 2 gt 0
x3 gt 0
X gt 0
x gt 2
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Graphing

• IMPORTANT Log functions are the inverses of
  exponential functions
• Inverse functions reflect about the line y
  x
• Easiest method for graphing without a calculator
• Write the exponential function yax
• Find its ordered pairs
• Switch the ordered pairs and graph the new points
• Shifts can occur just like exponential
  functions, but remember that the domain can be
  limited

Parent graph Domain (0, 8) Range (-8, 8)
X-intercept (1, 0) V.A. x 0 (y-axis)
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Ex Graph f(x) log2x

• Change to exponential function
• Create a table of values
• Flip the x and y values
• Graph the new set of values

f(x)log2x
f(x)2x
y2x
ylog2x