Definition of logarithms?

Question

logarithms of numbers!

Answer 1

Multiplication is a shortcut for addition, example : 3x5 = 5+5+5


Exponents are a shortcut for multiplication, example :
4 to power of 3 = 4x4x4


Logarithm is a shortcut for exponents.

Answer 2

Logarithms is when the Multiplication is a shortcut for addition, e.g. 6 X 5 = 5+5+5+5+5+5 and also logarithm is a shortcut for exponents e.g. Exponents are a shortcut for multiplication, e.g.
6 to power of 3 = 4x4x4x4x4x4

Answer 3

Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. The logarithms which they invented differed from each other and from the common and natural logarithms now in use. Napier's logarithms were published in 1614; Burgi's logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. Napier's approach was algebraic and Burgi's approach was geometric. Neither men had a concept of a logarithmic base. Napier defined logarithms as a ratio of two distances in a geometric form, as opposed to the current definition of logarithms as exponents. The possibility of defining logarithms as exponents was recognized by John Wallis in 1685 and by Johann Bernoulli in 1694.

The invention of the common system of logarithms is due to the combined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier's original logarithms. Their real significance was not recognized until later. The earliest natural logarithms occur in 1618.

Logarithms are useful in many fields from finance to astronomy.

Shortcuts

Multiplication is a shortcut for addition. Recall that means 5 + 5+ 5. Exponents are a shortcut for multiplication. Recall that means . Logarithm is a shortcut for exponents.

Answer 4

The logarithm is the mathematical operation that is the inverse of exponentiation (raising a constant, the base, to a power).

The logarithm of a number x in base b is the number n such that x = bn.

Answer 5

The logarithm of a number is the power to which the base must be raised to give that number.

So the log of 10 (to base 10) is 1

The log (to base 2) of 8 is 3

....and so on

Answer 6

I'll stick to logs to the base 10 (as opposed to base x) as it will be easier to understand;
1= 10 to the power of 0
10 = 10 to the power of 1
100 = 10 to the power of 2
1000 = 10 to the power of 3 etc

So log 1 = 0
log 10 = 1
log 100 = 2
log 1000 = 3... in other words the power you have to raise 10 to to get that number.
For numbers between these values you will have to use a calculator or log tables but 5 will be 0.something
14 will be 1.something
Actual values : log 5 = 0.69897
log 14 =1.146128
log 657 = 2.8178641.
Is it still as clear as mud?? Sorry,I tried.

Answer 7

let log represent the logarithm of
a number and k be the base
number

then,

log(x)(to to a base k) is the power
to which k must be raised in
order to equal x
that is

y=log(x)(to to a base k)

raise k to the power of each side

k^(y)=k^(log(x)(to the base k))
=x

example:
let k=8 and x=64
therefore,y=log(64)(to the base8)
=2
check;
k^(y)=k^(log(x)(to the base k))
8^2 =8^(log(64)(to the base 8))
64 =8^2=64

the most common logarithm bases
are 10 and e {the natural number}

i hope that this helps

Answer 8

Thanks to my own bad behaviour and my maths teacher of almost 30 years ago deciding 'lines' was the best form of punishment.

"You boy, shall write out the definition of logarithm 100 times", I can still chant

The logarithm of a number to a given base, is the index (or the power) by which the base must be raised to give the given number

If we break that down

The logarithm of a number (lets say 100)

To a given base (lets use base 10 just now 'cos its easy)

is the index (or the power) by which the base must be raised: that is you must raise 10 to the power of 2 "to give the given number" which was of course 100

I,e the logarithm of 100 to the base 10 is 2

Some examples

log of 16 to the base 2 is 4 because 2 to the power of 4 is 16.

Ln (natural log) of 37 is 3.61 because you have to raise 2.781828 (the base for natural logs) to the power of 3.61.... to get 37

And so on

Answer 9

Logarithms replace the process of multiplying by "Looking up in a table", adding, and "looking up the answer" in the same tables. In the seventeenth and eighteenth centuries when navigation at sea became practical with Harrisons Chronometer and a sextant to find the sun and star positions a method of simple accurate multiplication (and division) was needed to convert time and astronomical measurements to a position on the chart.The education of the average ships master was not up to multiplying long numbers, but he probably could add up, so tables of logarithms were prepared by Napier in the first instance. As an interesting aside, subsequent publishers of log tables allegedly make a couple of known small errors somewhere to mark the tables as their own. Anyone copying the tables has to check every number before adding their own errors, so they might as well do the job properly and recalculate everything. Nowadays we use calculators, computers, GPS instead, which is fine till the batteries die

Answer 10

The logarithm is the mathematical operation that is the inverse of exponentiation (raising a constant, the base, to a power).

The logarithm of a number x in base b is the number n such that x = b^n.

Answer 11

The logarithm is the mathematical operation that is the inverse of exponentiation (raising a constant, the base, to a power).

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