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Title: Logarithms


1
Logarithms
2
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.

3
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

4
Example 1
  • 10log x X
  • 10 to the is also the anti-log (opposite)

5
  • Log 23.5 1.371
  • Antilog 1.371 23.5 101.371

6
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.

7
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

8
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

9
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.

10
Problem 1
  • log (x)2 log 10 - 3 0

11
Solution
Try It Out Problem 1 Solution
                                                  
                                               
                                        
12
Problem 2
  • 3.5 ln 5x

13
  • Get rid of the ln by anti ln (ex)
  • e3.5 eln 5x
  • e3.5 5x
  • 33.1 5x
  • 6.62 x

14
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

15
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

16
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
17
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
18
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

19
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?

20
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!

21
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

22
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

23
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

24
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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