Logarithms

In the days before mechanical and electronic calculators, arithmetic had to be performed by hand.  Astronomers and other scientists spent much of their lives making arithmetic computations.   Since multiplication and division were more time-consuming than addition and subtraction, mathematicians began to think about ways to simplify these 2 arithmetic operations.   In 1614, the Scotsman, John Napier, invented logarithms ("proportionate numbers", in Greek) to save computational time.   He devised his logarithmic tables using geometric logic so that a logarithm corresponded to any positive real number.   As a result, all multiplication could be reduced to addition and all division reduced to subtraction.   Napier did not invent the concept of a logarithmic base (see below); nor did he relate logarithms to exponents (see below), which came later from other mathematicians.

In 1622, an English mathematician, William Oughtred, invented the slide-rule for mutliplying and dividing numbers based on logarithms.   The slide rule could be used in place of logarithmic tables.   Up to the 1950's, when mechanical calculators were used mainly by accountants, and electronic calculators had not yet been invented, engineers and scientists used log tables and slide rules to perform calculations.   Mechanical calculators and slide rules made calculating more rapid, but they still took more much more time than calculating by electronic calculators and computers.

Recall that multiplication, represented by the symbol x or •, is successive addition: 3 x 2 = (1 + 1 + 1) + (1 + 1 + 1) and powers are successive multiplications: 2³ = 2 x 2 x 2 = (1 + 1) x (1 + 1) x (1 +1).

Definition of an exponential function:

If b > 0 and b not= 1 (b is any positive real number other than 1), then y = b^x is called an exponential function.   b is called the base of the function.   x can be any real number and y > 0; that is, logarithms are defined only for positive numbers.   This limitation can be surmounted somewhat by the use of complex numbers, but that is beyond the scope of this discussion.

For any exponential function, when b > 1, then y increases as x increases.   This is called a growth function.   When b < 1, y decreases as x increases.   This is called a decay function.   Both are exponential functions.

Two common bases are b = 10 and b = e (= 2.71828... , an irrational number, as explained below):

A growth function with base 10 takes the form, y = 10^x, and a growth function with base e takes the form, y = e^x.

The modern definition of a logarithm ("log", for short) is based on exponents.

Since 8 x 16 = 2³ x 2⁴ = 2³⁺⁴ = 2⁷ = 128.   Likewise, 8 / 16 = 2³ x 2^-4 = 2^3-4 = 2^-1 = 1/2.   In other words, if numbers can be converted to a common base, one need only add the exponents to determine the result.   This idea can be extended to any positive number.   Thus, since 8 = 2³ and 16 = 2⁴, then the number 11 with the same base 2 must have an exponent somewhere between 3 and 4.   Therefore, if we had a table of exponents for all numbers to some convenient base, then multiplication would be reduced to addition and division would be reduced to subtraction.   Almost any base other than 1 will suffice for logarithmic tables, but it turns out that base 10 and base e are the most convenient.

For example, the log of 11 to the base 10 is given by the expression, 11 = 10^a, where a, the exponent of 10, is called "the logarithm of 11", which is 1.04139.   In other words, 10 raised to the 1.04139 power equals 11.   This can be verified on any calculator, but in the "good old days", it had to be obtained from a log table.   (Maybe the good old days were not so good in every respect.)

Definition of a logarithm:

If b > 0 and b not= 1 and y > 0, then

The above 2 equations show the relationship between logs and exponents.   In words, the logarithm of y is the exponent x.

Thus, 4³ = 64 and therefore, log₄64 = 3.   Also, 5^-2 = 1 / 25 means the log₅ (1 / 25) = -2.

Definition of a logarithmic function:

If b > 0 and b not= 1, then y = log_bx.

The logarithmic function is the inverse of the exponential function. Also, an exponential function translated to a logarithmic scale will appear linear when graphed.

The value of logarithms as aids to simplifying calculations comes from these two simple relationships, which can be proved:

The above two relationships were used in the past to add and subtract rather than multiply and divide.

Also, to simplify calculating exponents, especially when they are not decimal fractions,

To show the efficacy of logs in saving time, the old timers would have calculated 43.734 x 12.438 by looking up the log values of these numbers from a log table.   From a log table, the log of 43.734 = 1.6408 and the log of 12.438 = 1.0948.   Therefore, the log 43.734 + log 12.438 = 1.6408 + 1.0948 = 2.7356.   Then, the antilog (inverse logarithm) would be looked up in the table opposite the log 2.7356 to find the answer to the problem.   Interpolation between table numbers was usually required.   In our sample problem, the antilog for 2.7356 is 544.00, or, more accurately with a calculator, 543.963, which may calculate logs to 9 or more places.   Log tables usually gave logs to decimal places to only 4 or 5 places, so the results with fewer places are not as accurate.   The decimal part of the log is called the mantissa.   The whole number part of the log is called the characteristic.

Note that the logs of numbers from 0 to 1 are negative, the logs of numbers from 1 to 10 is between 0 and 1, the logs of numbers from 10 to 100 is between 1 and 2, etc.   The logs of very small numbers close to zero (0) are very large.

The two most common bases are b = 10, called the common base, and b = e, where e is the natural base and equal to the irrational number, 2.71828... .   However, a log to any base can be converted to any other base (usually base 10 or base e) by this formula:

To avoid the inconvenience of writing the base numbers, common and natural logs are usually abbreviated as follows:

Note on e:   e = 2.71828... is the irrational number that is the sum of a special function that occurs frequently in higher mathematics:

Although the base e may appear to be cumbersome, in fact, it simplifies much theoretical math work.   For a simple example, it simplifies financial compound interest problems:

If A = the total $ amount in a bank after 1 year on a bank balance (principal), P, where the annual interest rate is r and the number of compounds per year is n, then

A = P•(1 + r / n)ⁿ = P•[(1 + r / n)^n/r]^r.

Letting k = n / r, this equation becomes:

A = P•[(1 + 1/k)^k]^r

As k gets large, the number inside the brackets approaches e, so that the complicated formula for A when P is compounded continuously, is A = P•e^r.   Over several years, t,   A = P•e^rt.   What was said for a bank is equally true for any financial investment, e.g., stocks, bonds, real estate.   In fact, the formula is true for any quantity increaed or decreased by a fixed percent, because that represents an exponential equation.   For example, the present value of equipment that decreases by a fixed percent each year can determined by a similar formula.

Although no one uses log tables and slide rules today to facilitate calculations, except for the mathematicians who program microprocessors and computers, logarithms and logarithmic functions are extremely valuable to summarize and quantify natural phenomena in science and engineering where growth or decay follows an exponential function.   Logs also are useful to summarize data that vary greatly in magnitude, e.g., number of molecules and energy levels.   A quantity 10x greater is 1 more on a log scale.   A quantity 100x greater is 2 more on a log scale.   A quantity 1000x greater is 3 more on a log scale.   Etc.   Therefore, logarithms represent highly compressed forms of large variations.   Examples of extremely large and extremely small numerals that are conveniently represented by logarithmic functions:

Bacteria cultures may be 3 x 10⁵ in size.   They propagate by mitosis and their propagation rate might be such that they double every few minutes, i.e., it is an exponential growth function.   For example, escherichia coli (e.coli) doubles every 15 minutes   It is convenient to express such large numbers with logarithms.

In seismology, the study of earthquakes, where energy ranges are enormous, the energy released by an earthquake can be conveniently described by the Richter Scale, which uses a logarithm:

R = 0.67•log(0.37•E) + 1.46, where E is the energy in kW x hours.   From this formula, when the energy released, E, increases by 31 times, R increases by 1.

Atmospheric pressure is approximately halved every 5 km increase in height above sea level.   This is a decay function that could be expressed by logarithms.

Radioactive carbon-14 has a half-life of 5,730 years and is used to determine the age of fossils.   All radioactive decay can be expressed conveniently by decay functions and logarithms.

The current value of plant equipment depreciating by a fixed percentage each year and the future value of financial investments appreciating by a fixed percentage each year can be expressed by logarithms and decay and growth functions.

A capacitor discharges with time, so that after 1 second, its charge might be 50% of its original value.   This is a decay function.

In chemistry, the pH of a solution determines the number of hydrogen ions (H⁺) in a solution, and thereby its acidity or basicity.   pH = log(1 / H⁺) = -log(H⁺).   Defining a neutral solution as pure water, which has H⁺ = 10^-7, then pH_water = 7.   A pH lower than 7 is acidic and a pH higher than 7 is basic.   A pH of 7 is neutral, i.e., neither acidic nor basic.   pH values range from 1 to 14, 1 to 7 are acidic and 7 to 14 are basic by definition.   Human gastric juices are a highly acidic pH = 2.   Acid rain has a pH of about 4.2, so it is about 630 times (10^7-4.2) as acidic as pure water.   This is a good example of how a logarithmic scale reduces extremely small numbers (10^-14 = .00000000000001) to a number more easily managed (14).

In acoustics, the human ear hears "loudness" or "softness" as a logarithm of the sound intensity (power).   This perception is approximated by the decibel:

In astronomy, the "brightness" of stars to the human eye follows a logarithmic function.

From the above examples, one can see that logarithms and logarithmic functions are part of the essential math that every scientist, engineer, and computer scientist must learn.