Understanding Logarithms

Traditionally logarithms were developed to simplify the mathematical operations of multiplication and division to the level of addition and subtraction. Logarithmic tables and slide rules have been superceded by the calculator in actual computational calculations, but as many natural processes are best described by logarithmic based equations, it is essential that students have an understanding of logarithmic scales and numbers. There are two common logarithms in use and both can be accessed through a scientific calculator. First, is the historical decimal base 10 scale, represented by the "log" and 10^X keys on your calculator (the later is also called the anti-log). Second, is the natural logarithm or base e scale, which is represented by the "ln" and e^x or exp(x) keys on your calculator. Although calculus is not a prerequisite to this class, you will be required to memorize many natural logarithmic relationships that have been derived from calculus. Thus it is imperative that you know the basic algebraic operations required to manipulate logs, and these are prerequisite competencies for this course. For those of you who have had calculus, this class will provide numerous opportunities to see how calculus can be applied to a natural science.

Lets start from the beginning.

Addition takes two or more numbers and adds their values to form a third.

2+2=4

We can repeat this process:

2+2+2=6

2+2+2+2=8

Multiplication allows us to represent this repetition as a number added to itself n times.

Ie.,

2x2 = 2+2 = 4

2x3 = 2+2+2 = 6

2x4 = 2+2+2+2 = 8

2xn

repeats this process n times.

Division represents the reciprocal of this operation, that is, if I take a number and divide by 2, I can determine how many times I had to add two to get that number.

8/2=4

so

2x4=8

or

2+2+2+2=8

note; if I take any number (Z) and multiply and divide the number by any other equivalent number (n), I get that number back, so they are inverse operations:

(Z x n)/n=Z

Exponentiation represents the repetition of multiplication as multiplication represents the repetition of addition

2x2 = 2² = 4

2x2x2 = 2³ = 8

2x2x2x2 = 2⁴ = 16

2x2x2x2x2 = 2⁵ = 32

2x2x2x2x2x2 = 2⁶ = 64

logarithms represent the inverse of this operation. In general, we can use the expression

log_b(X)=n

where b is a real number called the base of the log (the number being multiplied by itself) n times to get X.

If I take the log of a base 2 number, I can determine how many times I multiplied 2 to get that number

log₂64=6

so

2⁶=64

So we use logarithms if we want to know the value of n for which 2ⁿ = 32, in essence, we are asking for what value of n does 2 multiplied by itself n times give us 32?

log₂32 = 5

so

2⁵=32

Thus, exponentiation is often called the anti log.

If Y=log_b X

then X is the antilog_b of Y,

or X=b^Y

As this is the inverse of exponentiation

x = log10^x = ln(e^x)

Note in the previous equation we used log(x) to represent log₁₀(x) and ln(x) to represent log_e(x). The convention is to only indicate the base if it is neither decimal or natural e.

How does this reduce the process of multiplication and division to those of addition and subtraction? What is the value of

4x8=?

From above we would determine this is 2²x2³. Look at the exponents, this is

2²⁺³=2⁵=32

So we added exponents to perform multiplication.

Likewise, we could have taken the log₂ of the two numbers and added them, and then taken the antilog of the sum

log₂(4) = 2

log₂(8) = 3

2+3=5

2⁵=32

From this point we will focus on the decimal and natural based logs. It is important to understand these rules apply to logs and antilogs of any base. Of course since we have calculators, we no longer need to multiply via logarithms. But we do need to be able to manipulate equations to solve for unknowns.

Now lets review common logarithmic & exponential operations you will be required to know this semester. Although I have represented the logarithmic operations on base 10 and e, it is imperative you understand these operations work on any base. If you do not understand one of these, use a simple base 10 operation to convince yourself these are correct.

For example if a^maⁿ = a^(m+n) makes no sense to you, try it with base 10.

This is saying: 10²x10³ = 10⁽²⁺³⁾ = 10⁵ = 100,000

(ie., a hundred times a thousand is a hundred thousand)

You are required to know these operations.

Exponential Operations

Natural & Decimal based Logarithmic Operations

Lets apply these to the two state Arrhenius equation which relates the kinetic rate constant to the temperature. If a reaction has an activation energy of 40 kJ/mol at room temperature (20^oC), at what temperature would the rate double if all other variables were kept constant. That is, at what temperature (T₂) does k₂= 2k₁

Dividing the first state by k_2:

Substituting for k₂

Taking the natural log of both sides:

You should be able to recognize how this is the same equation as equation 15.7 of your text.