logarithms. logarithms to various bases: red is to base e, green is to base 10, and purple is to...
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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 exponentlog 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 (a+b)
• 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
Try It Out Problem 1 Solution
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 summaryNumber 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?
What’s 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.)
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