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

• Logarithms, or "logs", are a simple way of
  expressing numbers in terms of a single base.
• Common logs are done with base ten, but some logs
  ("natural" logs) are done with the constant "e"
  as their base.
• The log of any number is the power to which the
  base must be raised to give that number.

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• In other words, log(10) is 1 and log(100) is 2
  (because 102 100).
• Logs can easily be found for either base on your
  calculator. Usually there are two different
  buttons, one saying "log", which is base ten, and
  one saying "ln", which is a natural log, base e.
  It is always assumed, unless otherwise stated,
  that "log" means log10.

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Chem?

• Logs are commonly used in chemistry.
• The most prominent example is the pH scale.
• The pH of a solution is the -log(H), where
  square brackets mean concentration.

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Review Log rules

• Logc (am) m logc(a)
• Example log2 X 8
• 28 X
• X 256
• 10log x X
• 10 to the is also the anti-log (opposite)

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Example 2 Review Log rules

• Example 2 log X 0.25
• Raise both side to the power of 10
• 10log x 100.25
• X 1.78

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Example 3 Review Log Rules

• 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

• The log of one number times the log of another
  number is equal to the log of the first plus the
  second number.
• Similarly, the log of one number divided by the
  log of another number is equal to the log of the
  first number minus the second.
• This holds true as long as the logs have the same
  base.

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Multiplying and Dividing logs

• Log (a b) log a log b
• Log (a / b) log a log b

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Try It Out Problem 1 Solution log (x)2 log 10
- 3 0
Try It Out Problem 1 Solution
                                                  
                                               
                                        
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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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Solve the following problem. 7 ln5x
ln(7x-2x)

• Simplify! 7 ln 5x ln 5x (PEMDAS)
• Log (ln) rules 7 2 ln 5x Adding goes to mult.
  when you remove an ln.
• (7 / 2) ln 5x
• 3.5 ln 5x
• 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

• Negative powers of 10 may be fitted into the
  system of 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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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 log x1/y (1/y )log x ln ln x1/y (1/y)ln x
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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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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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Sig Figs and logs

• For any log, the number to the left of the
  decimal point is called the characteristic, and
  the number to the right of the decimal point is
  called the mantissa.
• The characteristic only locates the decimal point
  of the number, so it is usually not included when
  determining the number of significant figures.
• The mantissa has as many significant figures as
  the number whose log was found.

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SHOW ME!

• log 5.43 x 1010 10.735
• The number has 3 significant figures, but its log
  ends up with 5 significant figures, since the
  mantissa has 3 and the characteristic has 2.
• ALWAYS ASK THE MANTISSA!

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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 15.18922614... 15.189
• 3 lots
  mantissa of 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.)

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QED

• Question?