Chapter 2

Background Mathematical Material

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Logarithms

If N = bx then x = logb N

For example, 100 = 102 thus log (100) = 2 [base 10 assumed] or 100 is the antilog of 2.

These are called common logs (log). Natural logs (ln) use the base e (=2.7183). Note the use of log and ln to denote common or natural logarithms, respectively.

To convert from common log (base 10) to natural log (base e)

use 2.303 x log10(N) = lne(N)

that is, ln(10) x log(N) = ln(N)

For example, ln (100) = 2.303 x log (100) = 2.303 x 2 = 4.606


Common logs are often used with equilibrium equations and buffer or pH calculations. Logarithms to base e are often used in pharmacokinetics and other kinetic processes.

Before calculators, logarithms were used to multiply or divide numbers. The two numbers to be multiplied or divided would be converted to logarithms. For multiplication the logs are added and for division the logs are subtracted.

Examples:
a) 23.7 x 56.4 = x

To find x take the natural log of both numbers, add and take the 'anti'-log (base e)

ln(23.7) + ln(56.4) = ln(x)

3.1655 + 4.0325 = 7.1980 = ln(x)

x = 1337

or 23.7 x 56.4 = e3.1655 x e4.0325 = e(3.1655 + 4.0325) = e7.1980 = 1337

b) 6.75 / 14.7 = y

To find y take the common log of both numbers, subtract and take the 'anti'-log (base 10)

log(6.75) - log(14.7) = log(y)

0.8293 - 1.1673 = -0.338 = log(y)

y = 0.4592


If you can find a slide rule you will notice that the scale on the slide is proportional to log of the number. By aligning numbers on the slide you can quickly add or subtract the length of each number and thus multiply or divide the numbers. When carefully preform the result can be quite accurate. Later we will look at semi-log graph paper. The y-axis on this paper also has a scale proportional to the log of the number.

Using semi-log graph paper as a slide rule for multiplication

Figure 2.3.1 Illustrates the Similarity of Semi-Log Graph Paper and a Slide Rule

Figure 2.3.1 illustrates the multiplication of 23.7 x 56.4 using a slide rule to give approximately 1300. The decimal point is determined by rough estimation.

Software: Caveman's Calculator by Brian Stephanik (Mac version)


Table 2.3.1 Table of log x and ln x Values
x log10x lnex
0.5 -0.301 -0.693
1 0.0 0.0
5 0.699 1.609
10 1.0 2.303
50 1.699 3.913
100 2.0 4.605

Plot of log x or ln x versus x

Figure 2.3.2 Plot of log x or ln x versus x

Click on the figure to view the interactive graph


References

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