Chapter 2 Background Mathematical Material

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Exponents

Definition:

N = b^x or N = b^x or N = b ** x
where N is the number, b is the base, often 10 but also e ( = 2.7183... Napier's constant), and x is the exponent (or power term when an integer as in 10² is 10 to the power 2).

Note use of ^ (common calculator or single line format) or ** (common computer language format) for general exponentiation. Exponents with base e may also be expressed as exp(x) [= e^x] (a commom computer function format).

With the same base, exponents can be added or subtracted

For example; a^x x a^y = a^(x+y) to perform multiplication

a^x / a^y = a^(x-y) to perform division.

Some Example Calculations

a) 100 = 10² = 10^2 = 10**2

b) 100 = e^4.605 = e^4.605 = e**4.605 where e = 2.7183 !!!

c) 10 x 100 = 10¹ x 10² = 10¹⁺² = 10³ = 1000

d) 10 x 100 = e^2.303 x e^4.605 = e^6.908 = 1000 when the base is the same you can add exponents to multiply numbers

Subtract exponents to divide

e) 5.6/1.2 = e^1.723 / e^0.182 = e^{1.723 - 0.182} = e^1.541 = 4.67

Use your calculator to check this answer

In pharmacokinetics and the study of other rate processes we are interested in numbers expressed as a base 10 or e with a negative exponent. Exponential decay refers to the decrease in the value of the number as the negative exponent increases in magnitude. Exponential decay is illustrated in tabular form and graphical form in Table 2.2.1 and Figure 2.2.1, respectively.

Table 2.2.1 Table of 10^-a*x or e^-a*x versus x

x
0.0

0.5

1.0

1.5

2.0

2.5

3.0

10^-x
1.000

0.316

0.100

0.032

0.010

0.003

0.001

e^-x
1.000

0.607

0.368

0.223

0.135

0.082

0.050

Figure 2.2.1 Linear plot of 10^-a*x or e^-a*x versus x

Click on the figure to view the interactive graph

Try your own calculation

References

Exponential function at Wikipedia
Exponential function at MathWorld
Search for Exponential function at Google

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*Table 2.2.1 Table of 10^-ax or e^-ax versus* x**
x	0.0	0.5	1.0	1.5	2.0	2.5	3.0
10^-x	1.000	0.316	0.100	0.032	0.010	0.003	0.001
e^-x	1.000	0.607	0.368	0.223	0.135	0.082	0.050

10^-x is:
e^-x is: