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Logarithms

The Essence of Logarithms

Logarithms have found its place in most scientific formulas and measures.Units expressed in logarithmic scales reduce wide-varying quantities intosmaller scopes. For instance, sound pressures and voltage ratios can be

quantified using a logarithmic unit called decibels. Also, the pH value is alogarithmic measure for the acidity of an aqueous solution. Moreover, itcan aid in forensic accounting, describes musical intervals, and appear inthe formula for estimating the number of prime numbers.

Brief History of Logarithms

In 1614, Scottish mathematician John Napier

published his discovery of logarithms. His mainintention was to assists the multiplication of

quantities that were calledsines. The whole sinewas the value of the side of a rightangled trianglewith a large hypotenuse.

After some time, a London professor, Henry Briggs(15611630) became interested in the logarithmic

tables prepared by Napier. Briggs traveled toScotland just to visit Napier and discussed the said

approach. They worked together in makingimprovements such as the base 10 logarithms.Later, Briggs developed a table of logarithms that

has remained in use until the advent of calculators

and computers. Common logarithms occasionallycalled Briggsian logarithm.

The present notion of logarithm is based on the

work of Leonhard Euler, who made its connectionwith exponential function during the 18

thcentury.

Indeed, the natural logarithm has the Eulersnumbere (2.716) as its base.

John Napier (15501617), the 7th

Laird of Merchiston, on his textA

Description of the Admirable Table

of Logarithm.

By shortening the labors, the

invention of logarithms doubled

the life of the astronomers.

Pierre Laplace

The logarithm is an example of a transcendental function,

along with other functions such as the trigonometric functions,hyperbolic functions, and their corresponding inverses.

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Definition of Logarithm

The logarithmic function with base b is the inverse of the

exponential function with base b. Alternatively,logby x if and only if

yx b

Note that the notation logb is use to denote the logarithm with base b.Also, it is timely to compare the domain and range of the logarithmic

function with its inverse.

Exponential Function Logarithmic Function

Domain 0x Range 0y

Properties of Logarithms

Given 0 and 1,b b then Example

1. log 1,b b 5log 5 1; log 1 2. log log ,nb ba n a 52 2log 3 5log 3

log ,nb b n 3

2 2log 8 log 2 3

3. If 0 and 0,x y thenlog log log ,b b bxy x y 2 2 2log 5 3 log 5 log 3

4.

If 0 and 0,x y then

log log log ,b b bx

x yy

2 2 27

log log 7 log 1111

5. If 0 and 1,m m thenlog

log ,log

mb

m

aa

b 102

10

log 9log 9

log 2

6. If 0 and 1a a , thenlogaxa x, 5

log 3 25 3 2

xx

The converse of each of the properties given above also holds true.

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Math Drill: Logarithms 1

A. Express each of the following exponential functions in logarithmic form.1. 5 12x 2. 4 73 9y 3. 2 110 x N

B. Express each of the following logarithmic functions in exponential form.4. 23log 9m

5. 26log 3 2y y

6. 9log 27 2 1x

C. Express the following logarithms in terms of the logarithms of thevariables x, y, and z, where the variables represent positive numbers.

7. 3log 7xyz

8. 2log3

y

xz

9. 2 35 64log x yz

10.2 13 4

32

981log x y

z

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D. Write the following logarithms in terms of a single logarithm, wherethe variables represent positive numbers.

11. log 2 log 3log 5logc c c cx y z

12. 31 2 16 6 6 62 3 4 5log 2 log log logx y z

13. 31 10 10 102 4log 3log log 4x y z

E. Using the logarithms: log 2 0.3010, log 3 0.4771, log7 0.8451, determine the value of each of the following logarithms.

14. log15

15. 10log 0.21

16. 510 249log 36

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Exercises: Logarithms 1

A. Change each of the following into logarithmic form.1. 3416 8 3. 3 54 32x 2. 32 1

34349

4. 121 5x

B. Change each of the following into exponential form.1. 2125 3log 25 3. 381 4log 27 2.

2394

log 2 4. 518 32 3log

C. Use the laws of logarithms, if applicable, to change each of theexpressions to sums and difference of multiples of logarithms.

1. 4 33log 5 x z 3. 5 2 43log zx y

2. 27 1 3 2log5 1

x x

x x

4.

5243 3 5

log7 2 1

a

x x

x

D. Write each expression as a single logarithm.1. log loga ax y x y 2. 22 2 22 5 3 3 1log log2 3 3 2 1

x x x

x x x

3. 2 28 82 22 5 3 6 7 2log log12 3 10 8x x x xx x x x 4. 5 6 3 7 2 3 5 2 43 3 3 32 3 4 2 5 6 7log 2log 3log 4logx y y z w x w y zw z w x y z x

E. Given the following logarithms: alog x 1.74787, alog y 1.51314, alog x y 2, and alog x y 1.23227. Evaluate the following

logarithms.

1. 2 2loga x y 3. 1 1loga x y 2.

log

yx

a y x 4. 2 2

1 1

loga x y