In the examples above the formula the base is 10. Further examples using base 10, base 2, and base e (where e = 2.718...) are given below in Table 1.
 

The first four entries in the base-10 section look natural as do the entries in the base 2, but few students would immediately guess .301 as the appropriate exponent for 2 = 10^x. Further, the natural base e (e = 2.71828..) probably seems at first an illogical base for representing numbers. As will be discussed below, exponential functions of the type y = ae^bx are very common in describing physical and chemical systems, and a basic understanding of this type of function is necessary.

The mathematical statement N = B^x serves as the basis for defining logarithms. The logarithm of a number (N) in base (B) is defined as (x).

            logarithm_(B)N = x

Two bases, 10 and e, are in common usage. The shorthand representations are:

base 10 logarithm₁₀ N = log₁₀ N = log N
and
base e ("Natural" logarithms) logarithm_e N = ln N

NOTE:
Only the numbers to the right of the decimal point in a logarithm are significant figures. The number to the left of the decimal point simply, in effect, tells us where the decimal point is and is not considered a significant figure.

There are a few basic rules for handling logarithms. Examples are given in base 10 but the rules are applicable to any base.

Rule 1:         log (a x b) = log a + log b
Examples: 
 

log (2000) = log (2 x 1000) = log 2 + log 1000 = .301 + 3 = 3.301
log (.004) = log (4 x .001) = log 4 + log .001  = .602 - 3 = -2.398

log (2/4) = log 2 - log 4 = .301 - .602 = -.301
log (2/.4) = log 2 - log .4 = log 2 - [log (4 x .1)] = .301 - [.602 - 1] = .699

log (4)³ = 3 log 4 = 3 x .602 = 1.806
log (8)^1/2 = 1/2 log 8 = 1/2 x .903 = .452

log 10^-8 = -8
ln e⁴ = 4

Examples: 
 

antilog .301 = 10^.301 = 2
antilog 1.806 = 10^1.806 = 64
antilog -2.398 = 10^-2.398 = .004
antiln 2.303 = e^2.303 = 10
antiln 1.386 = e^1.386 = 4

• INV + log or INV + ln

or
• 10^x or e^x

or
• y^x where y = BASE.

One of the very common functional relationships appearing from experimental observations is that of an exponential increase or decrease. This takes the form of an expression as

In the more general form this appears as

where a groups together any quantities that can be considered to be constant during the variation of x and y. When graphed, these functions appear as seen in Figure 1 (note that when x = 0, y becomes = 1). Base e logarithms can be converted to base 10 logarithms.

Since it is very difficult to obtain the exact form of the coefficient a from graphs such as these, it becomes more convenient to apply logarithms to functions in this form. Thus,
y = e^ax becomes ln y = ax; y = 10^a'x becomes log y = a'x

and a plot of ln (or log) y versus x will then give a straight line whose slope will be "a." The same considerations apply to these expressions when they are straight lines as to any straight line.

In a simple case, consider the data from Table 3 for the decomposition of hydrogen peroxide. Two ways in which these data can be graphed are shown below: (a) concentration against time on rectangular coordinates, (b) log concentration against time on rectangular coordinates. These are illustrated in Figures 2a - 2b.
 

Whether one uses the logarithmic or exponential form for such relationships depends on what is to be determined. Mathematically the exponential form has certain advantages while graphically the logarithmic form is more informative. Any functional relationship in this form has recognizable characteristics that are more obvious when one more symbol is included in the general expression. Thus,

where y_o is a constant determined by the fact that when x = 0, the value of e^-ax = 1 and under these conditions, y = y_o. The following examples illustrate the general characteristics of this type of function. The "a" is a specific constant in each example.

1. The decrease in the concentration of reacting material c per unit time t is proportional to the concentration and is expressed as

            c = c_oe^-at
where c_o is the concentration at time t = 0 (at the beginning of the experiment).
2. The decrease in the intensity of the incident light, I, is proportional to the depth, l, of the absorbing material and is expressed as

        I = I_oe^-al
where I_o is the intensity of the incident radiation at l = 0, before it passes through the absorbing material.
3. The pressure of the atmosphere P decreases at a rate proportional to the altitude h and is expressed as

        P = P_oe^-ah
where P_o is the pressure at h = 0, sea level.

1. Plot the given numerical values for x and y on a piece of graph paper using y as the vertical axis (ordinate) and x as the horizontal axis (abscissa). If a graphing program for a microcomputer is available, it provides a convenient way go examine a graph without resorting to a paper copy. The experimentally independent value is generally put on the x axis, and the experimentally measured, dependent, variable on the y axis.
2. If the resulting plot shows a straight line, the equation for the original numerical values is easily determined since the equation of a straight line is

        y = mx + b
where m = slope and b = intercept of line with y axis (at x=0).

Figure A. Positive slope 
slope = m = rise/run = 2/3 

 

Figure B. Negative slope
slope = m = rise/run = -2/2 = -1

Calculate or read off the values for m and b, then insert the m and b values into the equation y = mx + b. The result should be the equation you're looking for. To double check your work, try some of the original given values for x in your equation and see if y comes out as expected. The equation of the line in Fig. A is y = 2x/3 + 5/3 and for Fig. B is y = -x + 5 = 5 - x.
 

1. If the plot resulting from step I is not a straight line, you are not particularly likely to happen on the equation by inspection. I.e. the fact that if x = 3, y = 9, (a single point) does not imply that the equation is y = x² (holds for all points). One method is to convert the original data to a straight line. In this course we will encounter several relationships of the form

        y = ae^-b/T
where y is some property of the system, a and b are constants and T is the temperature in Kelvins. These functions are all linearized by taking the natural logarithym of the expression.

ln y = ln a - b/T. A plot of ln y vs 1/T will give a straight line with slope -b and intercept ln a.

  (a) y = 3x² + 4
  (b) y = 3e^2x
  (c) y = 6e^-4x
  (d) y = 3e^4/x
  (e) y = 2e^-8/x

D = -2.5 x 10^-3 T + 13.60