Logarithms and Their Graphs

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Logarithms and Their Graphs
? John Napier (creator of logarithms)

• By Jesus Rocha
• Period 2 Pre-Calculus

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Base b in Logarithm Problems

• The logarithm to the base b of x, log x, is the
  power to which you need to raise b in order to
  get x. log x y     means     b x
• Logarithmic Form Exponential form
• Rules
• 1. Log x is only defined if b and x are both
  positive, and b ?1
• 2. Log x is called the common logarithm of x,
  and is sometimes written as log 10.
• 3. Log x is called the natural logarithm of x

b
b
y
b
10
e
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Solving Logarithms
If log 1,000 3 (or the logarithm to the base
10 of 1,000 is 3) then its exponential form would
be 10 1,000 Solving - Move base 10 to the
left of log (10 log 1,000 3) - It is easy to
figure out that 10 to the power of 3 equals 1,000
so the exponential form would be written as 10
1,000
10
3
3
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Laws of Logarithms

• If the logs are being asked to be multiplied, log
  x (mn), then you should add the Logs log m
  log n
• ex log (4x8) log (4) log (8) 235
• If the logs are being asked to be divided, log
  (m/n), then you should subtract the Logs log m
  log n
• ex log (8/4) log (8) log (4) 3-21
• 3. b 1

b
b
b
2
2
2
b
b
b
2
2
2
0
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Graphing Logarithms
By nature of the logarithms, most log graphs tend
to have the same shape, looking similar to a
square root graph
Square Root Graph
Logarithm Graph
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It is simple to graph exponentials. For instance,
to graph y 2x, you would just plug in some
values for x, compute the corresponding y-values,
and plot the points. A negative number or 0 would
make it a little more difficult to solve -
Since 20 1, then log (1) 0, so (1, 0) is on
the graph - Since 21 2, then log (2) 1, so
(2, 1) is on the graph - Since 22 4, then log
(4) 2, so (4, 2) is on the graph - Since 23
8, then log (8) 3, so (8, 3) is on the
graph - Since 3, 5, 6, and 7 arent powers of 2,
they wouldnt work well with each other
2
2
2
2
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The Results