Logarithm for Chemistry Calculations

In chemistry especially in kinetics, thermodynamics, adsorption, photochemistry, electrochemistry, UV, nuclear chemistry, quantum chemistry, and statistical chemistry etc log is very frequently used tool to solve calculations. However, for users, applications of mathematical equation and similar others, unnecessarily cause phobia about a feeling that log and mathematics are difficult. Since the non-mathematical background people do not frequently use log and they also do not pay attention for understanding its simplicity so get unnecessary fear. This is very simple fundamental of mathematics and saves a lot of times. For example, number 1000 is depicted as 103 = 1000. For log form, the number is 1000, the base is 10 on which power is 3. So log101000 = 3 as log10 =1. So putting log form into original form is as 1x1000 = 103, both the sides the base 10 is maintained. The log of 1000 to base 10 has power or exponent = 3. The reverse operation generates 1000 as number. The 3 is power, this operation is for positive real numbers.

Let 1000 = x number, and in general, the log of x to base b is depicted as logb(x), when the base is implicit then the depictation is as log(x). The exponent y is raised to the base b as under.

x = by, taking log the same is noted as y = logb(x)

The base 10 is for common use, and e for the natural log and 2 is used for the binary numbers. Keeping same base log its multiplication as under.

Log (x.y) = log (x) + log (y), for example, log (4 x 9) = log (4) + log (9)

The log of the product of two numbers is the sum of the log of those numbers. Similarly, log reduces division to subtraction by the formula.

Log (x/y) = log (x) – log (y)

The log of the quotient of two numbers is difference between the logs of those numbers. Thus the log has applications in diverse fields like statistics, chemistry, physics, astronomy, computer science, economics, music, and engineering.

Log of positive real numbers

The log is denoted as bx = y where log x is logb(y), when base b = 2, then log2(8) = 3. The number 8 is depicted as 23 = 2 x 2 x 2 = 8. So base 2 works well with number 8 and log is log2(8) = 3. Similarly the number 16 is depicted as 24 = 2 x 2 x 2 x2 = 16, and its log2(16) = 4. Thus with number the exponent varies.

Interestingly if number is ½ then it could be depicted as 2-1=1/21 = ½. Hence log is log2(1/2) = -1. Similarly if number is ¼ then log is log2(1/4) = 2-2= 1/22= ¼, hence log2(1/4) = -2.

Similarly the log2(3) ≈ 1.58.

Logarithm of a product

The log3(9 x 27) = log3(243) = 5 due to 35 = 243. Also the sum of log3(9) = 2 and log3(27) = 3 also equals 5. Basically, the log defines contributory variables of individual parameter of certain operation executed on specific materials like aqueous mixtures of proteins, polymers, amino acids, dendrimeres, binary, ternary, ionic or molecular mixtures. For example, the Friccohesity depends on viscous flow time t and pendant drop number n. The t and n are contributory variables to the friccohesity for constant liquid volume filled in, in Survismeter functional bulb is taken as base. Thus, the friccohesity could be depicted as logV(t x n) where frictional and cohesive forces do contribute a lot.

Log of a power

The same is noted as logb(xp) = p logb(x), example, log2(64) = log2(43) = 3 log2(4) = 3 x 2 = 6. Thus x number is generated as x = blogb(x) and xp = (blogb(x))p = bp · logb(x).

The (de)f = de · f, where d, e and f are positive real numbers. Thus, the logb of the left hand side, logb(xp), and of the right hand side, p · logb(x), agree.

Change of base

The ba = x, thus the logk(x) = logk(ba) = a  logk(b).

The general restriction b ≠ 1 implies logkb ≠ 0, since b0 = 1.

Natural log is log to base e, the e is an irrational constant and equal to 2.7183. The natural log is noted as ln(x) or loge(x), when base e is implicit then as log(x). The natural log of x is ln(x)). Here the power to which e would have to be raised to equal x. For example, ln(7.389) is 2, because e2=7.389. The natural log of e itself (ln(e)) = 1 as e1 = e while the natural log of 1 (ln(1))=0 because the e0 = 1.