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.)