Chapter 2

Background Mathematical Material

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Exponents

Definition:

N = bx 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) [= ex] (a commom computer function format).

With the same base, exponents can be added or subtracted

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

or

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

Some Example Calculations

a) 100 = 102 = 10^2 = 10**2

b) 100 = e4.605 = 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 = e2.303 x e4.605 = e6.908 = 1000 when the base is the same you can add exponents to multiply numbers

Subtract exponents to divide

e) 5.6/1.2 = e1.723 / e0.182 = e1.723 - 0.182 = e1.541 = 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
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


Linear plot of 10^x or e^x versus x

Figure 2.2.1 Linear plot of 10-x or e-x versus x

Click on the figure to view the interactive graph


Try your own calculation

Calculator 2.2.1 Calculate 10-x or e-x

Enter your own values of x

10-x is:
e-x is:


References

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