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
• 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.
• 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.
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.
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)
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
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
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.
Multiplying and Dividing logs
• Log (a * b) = log a + log b
• Log (a / b) = log a – log b
Try It Out Problem 1 Solution
log (x)2 – log 10 - 3 = 0
Try It Out Problem 1 Solution
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
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
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
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
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!
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
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.
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!
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
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.)
QED
• Question?