There we discussed the need for extension of Exponents and introduction to the new branch of study called Logarithms.

It is a prerequisite here.

Proofs and Explanations for Formulas in Logarithms :

Proof and Explanation of Formula 1 : Properties of Logarithms

Formula from definition of Logarithm:

a^{x} = n ⇔ log_{a}n = x

This formula is directly from definition of Logarithm.

This formula from exponential form to logarithmic form and from logarithmic form to exponential form is useful in solving many a problem.

The student should be thoroughly familiar with it in applying in both directions.

In remembering this formula, the following two points are helpful.

Both in exponential form and logarithmic form, the base is same.

The exponent in exponential form is the value of the logarithm in logarithmic form.

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Proof and Explanation of Formula 2 : Properties of Logarithms

Logarithm as an Exponent to its Base:

a^{(loga n)} = n

Proof: From the definition of logarithm, we have if a^{x} = n..............(i) then x = log_{a}n ................(ii) Substituting the value of x from (ii) in (i), we get a^{(loga n)} = n (Proved.)

In remembering this formula, you have to observe that the base of the exponential form is same as the base of the logarithm in the exponent.

Proof and Explanation of Formula 3 : Properties of Logarithms

Logarithm of 1 to any Base:

log_{a} 1 = 0

Proof: From Laws of Exponents, we know a^{0} = 1 ⇒ log_{a} 1 = 0 (Proved.)

It is easy to remember Logarithm of 1 to any Base is zero.

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Proof and Explanation of Formula 4 : Properties of Logarithms

Logarithm of any number to the same Base:

log_{a}a = 1

Proof: From Laws of Exponents, we know a^{1} = a ⇒ log_{a}a = 1 (Proved.)

It is easy to remember Logarithm of any number to the same Base is one.

Proof and Explanation of Formula 5 : Properties of Logarithms

Logarithm of a Product:

log_{a} (mn) = log_{a}m + log_{a}n

Proof: Let log_{a}m = P ⇒ a^{P} = m ............(i) Let log_{a}n = Q ⇒ a^{Q} = n .............(ii) You might have observed that Equations (i) and (ii) are obtained by changing the Logarithmic form to Exponential form.

(i) x (ii) gives a^{P} x a^{Q} = mn ⇒ a^{P + Q} = mn ( Since a^{P} x a^{Q} = a^{P + Q} From laws of Exponents) ⇒ log_{a} (mn) = P + Q ( by changing Exponential form to Logarithmic form) By Replacing the values of P and Q, we get log_{a} (mn) = log_{a}m + log_{a}n (Proved.)

In proving this, see how we made use of changing Logarithmic to Exponential form and Exponential to Logarithmic form.

Remember that Logarithm of a Product is the sum of the Logarithms of the Factors of the Product.

Remember that the Formula is not for log(m + n) nor for log m x log n.

We have Formula for log (mn) and log m + log n.

We should be able to apply the formula from L.H.S. to R.H.S. and from R.H.S. to L.H.S.

Proofs and Explanations of Formula 6 to 10 : Properties of Logarithms

The following Link takes you to the Proofs and Explanations of Formula 6 to 10.

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