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Properties of logrithm | Logrithm properties|Vector studies|

July 13, 2022

 PROPERTIES OF LOGRITHM

  • PRODUCT PROPERTY

The product rule states that the multiplication of two or more logarithms with common bases is equal to adding the individual logarithms i.e.

log a (MN) = log a M + log a N

Proof

  • Let x = log aM and y = log a
  • Convert each of these equations to the exponential form.

⇒ a x = M⇒ a y = N

  • Multiply the exponential terms (M & N):

ax * ay = MN

  • Since the base is common, therefore, add the exponents:

a x + y = MN

  • Taking log with base ‘a’ on both sides.

log a (a x + y) = log a (MN)

  • Applying the power rule of a logarithmlog a Mn ⇒ n log a M

    (x + y) log a a = log a (MN)(x + y) = log a (MN)

    • Now, substitute the values of x and y in the equation we get above.

    log a M + log a N = log a (MN)

    Hence, proved

    log a (MN) = log a M + log a N

  • QUOTIENT PROPERTY

This rule states that the ratio of two logarithms with the same bases is equal to the difference of the logarithms i.e.

log a (M/N) = log a M – log a N

Proof

  • Let x = log aM and y = log a
  • Convert each of these equation to the exponential form.

⇒ a x = M

⇒ a y = N

  • Divide the exponential terms (M & N):

ax / ay = M/N

  • Since the base is common, therefore, subtract the exponents:

a x – y = M/N

  • Taking log with base ‘a’ on both sides.

log a (a x – y) = log a (M/N)

  • Applying the power rule of logarithm on both sides.

log a Mn ⇒ n log a M

(x – y) log a a = log a (M/N)

(x – y) = log a (M/N)

  • Now, substitute the values of x and y in the equation we get above.

log a M – log a N = log a (M/N)

Hence, proved

log a (M/N) = log a M – log a N

  • POWER PROPERTY

According to the power property of logarithm, the log of a number ‘M’ with exponent ‘n’ is equal to the product of exponent with a log of a number (without exponent) i.e.

log a M n = n log a M

Proof

  • Let,

x = log a M

  • Rewrite as an exponential equation.

a x = M

  • Take power ‘n’ on both sides of the equation.

(a x) n = M n

a xn = M n

  • Take log on both sides of the equation with the base a.

log a a xn = log a M n

  • log a a xn = log a M n ⇒ xn log a a = log a M n ⇒ xn = log a M n
  • Now, substitute the values of x and y in the equation we get above and simplify.

We know,

x = log a M   

So,  xn = log a M n ⇒ n log a M = log a M n

Hence, proved

log a M n = n log a M

BASE CHANGE PROPERTY

According to the change of base property of logarithm, we can rewrite a given logarithm as the ratio of two logarithms with any new base. It is given as:

log a M = log b M/ log b N

or

log a M = log b M × log N b

Its proof can be done using one to one property and power rule for logarithms.

Proof

  • Express each logarithm in exponential form by letting;

Let,

x = log N M

  • Convert it to exponential form,

M = N x

  • Apply one to one property.

log b N x = log b M

  • Applying the power rule.

x log b N = log b M


BY YATENDRA KUMAR
VECTOR STUDIES


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