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Logarithms

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.

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

Example 1

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

- Log 23.5 1.371
- Antilog 1.371 23.5 101.371

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.

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

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

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.

Problem 1

- log (x)2 log 10 - 3 0

Solution

Try It Out Problem 1 Solution

Problem 2

- 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

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

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

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

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

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

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?

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!

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

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

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

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.)