Logarithms

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Logarithms
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Logarithms

• Logarithms to various bases red is to base e,
  green is to base 10, and purple is to base 1.7.
• Each tick on the axes is one unit.
• Logarithms of all bases pass through the point
  (1, 0), because any number raised to the power 0
  is 1, and through the points (b, 1) for base b,
  because a number raised to the power 1 is itself.
  The curves approach the y-axis but do not reach
  it because of the singularity at x 0.

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Definition

• The log of any number is the power to which the
  base must be raised to give that number.
• log(10) is 1 and log(100) is 2 (because 102
  100).
• Example log2 X 8 28 X X 256

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Example 1

• 10log x X
• 10 to the is also the anti-log (opposite)

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• Log 23.5 1.371
• Antilog 1.371 23.5 101.371

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Logs used in Chem

• The most prominent example is the pH scale, but
  many formulas that we use require to work with
  log and ln.
• The pH of a solution is the -log(H), where
  square brackets mean concentration.

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Example 2 Review Log rules

• log X 0.25
• Raise both side to the power of 10 (or
  calculating the antilog)
• 10log x 100.25
• X 1.78

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Example 3 Review Log Rules

• Logc (am) m logc(a)
• Solve for x 3x 1000
• Log both sides to get rid of the exponent
• log 3x log 1000
• x log 3 log 1000
• x log 1000 / log 3
• x 6.29

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Multiplying and Dividing logs

• log a x log b log (ab)
• log a/b log (a-b)
• This holds true as long as the logs have the same
  base.

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Problem 1

• log (x)2 log 10 - 3 0

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Solution
Try It Out Problem 1 Solution
                                                  
                                               
                                        
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Problem 2

• 3.5 ln 5x

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• Get rid of the ln by anti ln (ex)
• e3.5 eln 5x
• e3.5 5x
• 33.1 5x
• 6.62 x

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Negative Logarithms

• We recall that 10-1 means 1/10, or the decimal
  fraction, 0.1.
• What is the logarithm of 0.1?
• SOLUTION 10-1 0.1 log 0.1 -1
• Likewise 10-2 0.01 log 0.01 -2

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Natural Logarithms

• The natural log of a number is the power to which
  e must be raised to equal the number. e 2.71828
• natural log of 10 2.303
• e2.303 10 ln 10 2.303
• e ln x x

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SUMMARY
Common Logarithm Natural Logarithm
log xy log x log y ln xy ln x ln y
log x/y log x - log y ln x/y ln x - ln y
log xy y log x ln xy y ln x
log x1/y (1/y )log x ln x1/y (1/y)ln x
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In summary
Number Exponential Expression Logarithm
1000 103 3
100 102 2
10 101 1
1 100 0
1/10 0.1 10-1 -1
1/100 0.01 10-2 -2
1/1000 0.001 10-3 -3
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Simplify the following expression log59 log23
log26

• We need to convert to Like bases (just like
  fraction) so we can add
• Convert to base 10 using the Change of base
  formula
• (log 9 / log 5) (log 3 / log 2) (log 6 / log
  2)
• Calculates out to be 5.535

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ln vs. log?

• Many equations used in chemistry were derived
  using calculus, and these often involved natural
  logarithms. The relationship between ln x and log
  x is
• ln x 2.303 log x
• Why 2.303?

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Whats with the 2.303

• Let's use x 10 and find out for ourselves.
• Rearranging, we have (ln 10)/(log 10) number.
• We can easily calculate that
• ln 10 2.302585093... or 2.303
• and log 10 1.
• So, substituting in we get 2.303 / 1 2.303.
  Voila!

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Sig Figs and logs

• For a measured quantity, the number of digits
  after the decimal point equals the number of sig
  fig in the original number
• 23.5 measured quantity ? 3 sig fig
• Log 23.5 1.371 3 sig fig after the decimal
  point

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More log sig fig examples

• log 2.7 x 10-8 -7.57 The number has 2
  significant figures, but its log ends up with 3
  significant figures.
• ln 3.95 x 106 15.189 the number has 5
• 3

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OK now how about the Chem.

• LOGS and Application to pH problems
• pH -log H
• What is the pH of an aqueous solution when the
  concentration of hydrogen ion is 5.0 x 10-4 M?
• pH -log H -log (5.0 x 10-4) - (-3.30)
• pH 3.30

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Inverse logs and pH

• pH -log H
• What is the concentration of the hydrogen ion
  concentration in an aqueous solution with pH
  13.22?
• pH -log H 13.22 log H -13.22 H
  inv log (-13.22) H 6.0 x 10-14 M (2 sig.
  fig.)