The document discusses logarithms and includes examples of solving logarithmic equations, graphing logarithmic functions, and properties of logarithms and the natural logarithm function. Key points covered include: - Logarithms are used to solve for exponents. - Common properties include loga1=0, logaa=1, and logaax=x. - Graphs of logarithmic functions f(x)=logax have a vertical asymptote at x=0 and an x-intercept at (1,0). - Examples demonstrate evaluating logarithmic expressions, solving equations, and using a logarithmic formula to estimate the age of an archaeological sample.

1. 3.2 Logarithms
• All of the material in this slideshow is
important, however the only slides than
must be included in your notes are the
Recap slides at the end.
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2. Intro
Solving for an Solving for a base Solving for an
answer exponent
1. 34 = x 2. x2 = 49 3. 3x = 729
81 = x x = ±7 log3729=3
New Calc:
Alpha window
5:logBASE(
Older Calc:
log(729)/log(3)
Answer: 6
b/c 36 = 729
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3. Now you try…
4. 25 = x 5. x3 = 125 6. 5x = 3125
log 5 3125 = x
Plug in calc
32 = x
3
x = 1253 3
log base ans = exp
x=5
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4. A logarithmic function is the inverse function
of an exponential function.
Every logarithmic equation has an equivalent
exponential form:
y = loga x is equivalent to x = a y
A logarithm is an exponent!
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5. Examples: Write the equivalent exponential equation
and solve for y.
Logarithmic Equivalent Solution
Equation Exponential
Equation
y = log216 16 = 2y 16 = 24 → y = 4
1 1 1
y = log2( ) = 2y = 2-1→ y = –1
2 2 2
y = log416 16 = 4y 16 = 42 → y = 2
y = log51 1=5y 1 = 50 → y = 0
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6. 3.2 Material
• Logarithms are used when solving for an exponent.
baseexp =ans  logbaseans=exp
• In your calculator: log is base 10 ln is base e
Properties of Logarithms and Ln
1. loga 1 = 0
2. logaa = 1
3. logaax = x
• The graph is the inverse of y=ax (reflected over y =x )
VA: x = 0 , x-intercept (1,0), increasing
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7. The graphs of logarithmic functions are similar for different
values of a.
f(x) = loga x
Graph of f (x) = loga x y-axis y y = ax y=x
vertical
asymptote y = log2 x
x-intercept (1, 0)
VA: x = 0
increasing x
x-intercept
(1, 0)
reflection of y = a x in y = x
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8. Example: Graph the common logarithm function f(x) = log10 x.
y
f(x) = log10 x
x
5
(1, 0) x-intercept
x=0
vertical –5
asymptote
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9. y y = ln x
The function defined by
f(x) = loge x = ln x x
5
is called the natural
logarithm function.
–5
y = ln x is equivalent to e y = x
Use a calculator to evaluate: ln 3, ln –2, ln 100
Function Value Keystrokes Display
ln 3 LN 3 ENTER 1.0986122
ln –2 LN –2 ENTER ERROR
ln 100 LN 100 ENTER 4.6051701
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10. Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
1
( )
ln  2  = ln e −2 = − 2 inverse property
e 
3 ln e = 3(1) = 3 property 2
ln 1 = 0 = 0 property 1
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11. −t
 1  8223
Example: The formulaR =  12 e (t in years) is used to estimate
 10 
the age of organic material. The ratio of carbon 14 to carbon 12
1
in a piece of charcoal found at an archaeological dig is R = 15 .
10
How old is it?  1  −t 1
 12  e 8223 = 15 original equation
10 
 −t 10
1
e 8223
= 3 multiply both sides by 1012
−t
10
1
ln e 8223
= ln take the ln of both sides
1000
−t 1
= ln inverse property
8223 1000
 1 
t = −8223  ln  ≈ −8223 ( − 6.907 ) = 56796
 1000 
To the nearest thousand years the charcoal is 57,000 years old.
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12. 3.2 Material Recap
• Logarithms are used when solving for an exponent.
logab=c ac =b
• In your calculator: log is base 10 ln is base e
Properties of Logarithms and Ln
1. loga 1 = 0  because when written in exponential form a0 = 1
2. logaa = 1  because when written in exponential form a1 = a
3. logaax = x because when written in exponential form ax = ax
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13. 3.2 Material Recap
Graph the function f(x) = log x.
y
f(x) = log10 x
x
5
(1, 0) x-intercept
x=0
vertical –5
asymptote
• The graph is the inverse of y=ax (reflected over y =x )
VA: x = 0 , x-intercept (1,0), increasing
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14. Practice Problems
Page 236 (all work and solutions on blackboard)
•#1–16 pick any 4
•17–22 pick any 3
•27–29 all
•31–37 pick any 4
•45–52 pick any 3
•53–59 pick any 3
•69–71 all
•79–86 pick any 4
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