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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
• 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.)