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Thought-Process to Discover Knowledge

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mathsExponentsFundamentals of Exponents

Logarithm : An inverse of Exponent

This page introduces "logarithm".

One of the inverses of exponent is logarithm. Logarithm is introduced with the following two.

 •  first principles -- Logarithm of a number is the power in the equivalent exponent.

 •  Simplified Procedure -- Logarithm of a number is found from prime-factorization of the number (if log evaluates to an integer). This introduction "logarithm is an inverse of exponent" is astoundingly clear and makes it simple for students.


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We learned that
 •  Subtraction is the inverse of addition,
 •  Division is the inverse of multiplication,

What is the inverse of "exponent"?

  • given result of exponentiation and base, find the power
  • given result of exponentiation and power, find the base
  • both the above
  • both the above

The answer is "both the above"

Two inverses are defined for exponents.

The exponent is not commutative.
`3^2=9` and `2^3=8`
`a^b !=b^a`

Exponent is
`(text(base))^(text(power)) = text(exp. result)`

if exponentiation result and power are given, then
`text(base) = root(text(power))(text(result)) = (text(exp. result))^(1/text(power))`

This inverse is called "root".

If exponentiation-result and base are given, then
`text(power) = log_text(base) (text(exp.result))`
This inverse is called "logarithm".

Which of the following is a meaning for the word "logarithm"?

  • a rhythm made using logs
  • numbers in ratio order
  • numbers in ratio order

The answer is "numbers in ratio order". The word logarithm is derived from original Greek words, "logos"( meaning ratio) and "arithmos" (meaning numbers).

logarithm is associated with a sequence increasing or decreasing in ratios, like `1, 10, 100, cdots`.

What is the term used to refer finding power from the exponentiation result?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is "logarithm".

Logarithm : Logarithm of a number to a given base of logarithm, is the power of the base that equals the number.

`log_2 8 = 3`
`2` is the base of logarithm
`8` is the number for which logarithm is calculated
`3` is the result of logarithm

`log_2 8 = 3` implies that `2^3=8`, and the operation logarithm finds the power in the exponent.

Finding Logarithm (First Principles) : Logarithm of a number is the power in the equivalent exponent.

eg: `log_4 64` is seen as exponent `64 = 4^3`. The power is `3` and so `log_4 64 = 3`

Which of the following is to find the power from `7^2 = 49`?

  • `root(7)(49)`
  • `log_7 49`
  • `log_7 49`

The answer is "`log_7 49`"

find `log_5 (125)`

  • `3`
  • `3`
  • `5`

The answer is "`3`".

To find `log_5(125)`, represent the value in the given base.
`125 = 5xx5xx5 = 5^3`

By first principles, `log_5(125) = 3`

Find `log_6(36)`

  • `2`
  • `2`
  • `6`

The answer is "`2`".

To find `log_6(36)`, represent the value in the given base.
`36=6xx6 = 6^2`

By first principles, `log_6(36)= 2`

Finding Logarithms (Simplified Procedure) : To find logarithm of a number, express the number in the given base.

eg: `log_10 1000` `=log_10 (10xx10xx10)` `=3`

                            
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