Logarithmic Properties (2024)

Back to the Table of Contents

Numbers and their Application - Lesson 17

Lesson Overview

• The Definition of a Logarithm
• The Four Basic Properties of Logs
• The Slide Rule
• Applications of Logarithms
• Homework

The Definition of a Logarithm

A logarithm is an exponent.

Note, the above is not a definition, merely a pithy description.

Just as subtraction is the inverse operation of addition, and taking a square root is the inverse operation of squaring,exponentiation and logarithms are inverse operations. Finding an antilog is the inverse operation of finding a log,so is another name for exponentiation. However, historically, this was doneas a table lookup.Some history was given earlier andthe formal definition is repeated below, this time with restrictions.

y = logbx if and only if by = x, where x > 0, b > 0, and b 1.

As noted above, the base can be any positive number (except 1). However, two choices are most usual: 10 and e=2.718281828....Logs to the base 10 are often call common logs, whereaslogs to the base e are often call natural logs.Logs to the bases of 10 and e are now both fairly standard on most calculators.Often when taking a log, the base is arbitrary and does not needto be specified. However, at other times it is necessary and mustbe assumed or specified.

At the high school level only,log x consistantly means log10x. In college, especially in mathematics and physics,log x consistantly means logex. A popular notation (despised by some) is:ln x means logex.

To calculate logs to other bases, the change of base rule below (#4) should be used.It is only multiplication by a constant (1 / log_ab).

The Four Basic Properties of Logs

logb(xy) = logbx + logby. logb(x/y) = logbx - logby. logb(xn) = n logbx. logbx = logax / logab.

These four basic properties all follow directly from the fact that logs are exponents.In words, the first three can be remembered as:The log of a product is equal to the sum of the logs of the factors.The log of a quotient is equal to the difference between the logsof the numerator and demoninator.The log of a power is equal to the power times the log of the base.

Additional properties, some obvious, some not so obvious are listed below for reference. Number 6 is called the reciprocal property.

logb1 = 0. logbb = 1. logbb2 = 2. logbbx = x. blogbx = x. logab = 1/logba.

The Slide Rule

The invention of logs was followed quickly by the invention of the slide rule. Slide rules simplify multiplication and divisionby converting these operations into addition and subtraction.This is done by placing the numbers on a scale which is logarithmic.Given below are the logs of some small integers.

n	log10n	logen
1	0.000	0.000
2	0.301	0.693
3	0.477	1.099
4	0.602	1.386
5	0.699	1.609
6	0.778	1.792
7	0.845	1.946
8	0.903	2.079
9	0.954	2.197
10	1.000	2.303

From this we can readily verify such properties as: log 10 = log 2 + log 5and log 4 = 2 log 2. These are true for either base.In fact, the useful result of 10³ = 10001024 = 2¹⁰ can be readily seen as 10 log₁₀2 3.

The slide rule below is presented in a disassembled state to facilitate cutting.(Also, by putting it below, it will be at the bottom of page 3 and have blankpaper behind it.)The portion above slides in the center of the portion below and shouldbe printed, then cut out for demonstration purposes as follows.

1. Align the left 1 on the D scale with the 2 on the C scale. Observe thenumber above 4 of the D scale on the C scale. Since these numbersare laid out on a logorithmic scale, you have shown that log 2 + log 4 = log (2×4) = log 8. Circle that 8.
2. Align the right 1 on the D scale with the 4 on the C scale. Observe thenumber below the left 1 on the C scale. You have just shown that log 10 - log 4 = log 2.5. Circle that 2.5.
3. Align the D scale and A scale. The A scale is laid out similarily, except there are two cycles present. Observe the number just above the 9 on the D scale.You have just shown that 2 log 9 = log 9² = log 81. Circle that 81.
4. See how the K scale can be used to cube things.
5. Notice how the CI scale can also be used to divide.

Normally there is a cursor (the original meaning, not the kind blinking on the computer screen) present which allows one toget about three decimal places of accuracy, hence the term slide rule accuracy.

Applications of Logarithms

Logs are used in a variety of applications in sciences, some of the mostcommon are: measuring loudness (decibels), measureing earthquake intensity (Richter scale),radioactive decay, and acidity (pH= -log₁₀[H⁺]). They are essentialin mathematics to solve certain exponential-type problems.

Following, is an interesting problem which ties the quadratic formula, logarithms, and exponents together very neatly.

log(2x+2) + log x - log(12) = 0 Simplify the logarithms by combining them.
log(2x² + 2x) - log(12) = 0
log((2x² + 2x)/12) = 0
After dividing by 2, exponentiate both sides (the base b is arbitrary, since it was not specified above)!
(x² + x)/6 = b⁰
(x² + x)/6 = 1
x² + x = 6
x² + x - 6 = 0
(x + 3)(x - 2) = 0x {-3, 2}

Blank space so when printed with Mozilla (oops, no boxes) it is in back of the slide rule.

However x -3since the domain of log is only the positive reals. (b^x can never be a negative number with b > 0).

The next example (6.11#51) combines logarithms with simultaneous equations. It is also very convenient to introduce the concept of substitution, which is so useful in calculus.

log₉x + log_y8 = 2.
log_x9 + log₈y = 8/3.

Let u=log₉x and v=log₈y.By the reciprocal property above,1/u=log_x9 and 1/v=log_y8.

We can rewrite our equations now as:

u + 1/v = 2
1/u + v = 8/3Solving by substitution,u = 2 - 1/v, thus: 1/(2 - 1/v) + v = 8/3.
3(1 + 2v - 1) = 8(2 - 1/v)
6v² = 16v - 8.
6v² - 16v + 8 = 0.
3v² - 8v + 4 = 0.
To this we apply the quadratic formula and find that
v = (8 ± (64 - 48))/6.
= (8 ± 4)/6 or 2, 2/3.
Thus u = 3/2 or 1/2 or (u, v)={(3/2, 2), (1/2, 2/3)}
Thus (x, y) = {(27, 64), (3, 4)}

BACK	HOMEWORK	ACTIVITY	CONTINUE

• e-mail: calkins@andrews.edu
• voice/mail: 269 471-6629; BCM&S Smith Hall 105; Andrews University;
• fax/classroom: 269 471-6646; Smith Hall 100; Berrien Springs, MI, 49104-0140
• home: 269 473-2572; 610 N. Main St.; Berrien Springs, MI 49103-1013
• URL: http://www.andrews.edu/~calkins/math/webtexts/numb17.htm
• Copyright ©1999–2005, Keith G. Calkins. Revised on or after November 8, 2005.