Log Formulas Derivation

Log formulas are very useful for solving various mathematical problems and these formula are easily derived using laws of exponents. Now lets learn about the derivation of some log formulas in detail.

Derivation of Product Formula of Log

Product formula of log states that,

log_b (xy) = log_b x + log_b y

This is derived as,

Let take, log_b x = m and log_b y = n…(i)

Now using definition of logarithm,

x = b^m and y = bⁿ

⇒ x.y = b^m × bⁿ = b^{(m + n)} → {by a law of exponents, p^m × pⁿ = p^{(m + n)}}

⇒ x.y = b^{(m + n)}

Converting into logarithm form,

m + n = log_b xy

from eq. (1)

log_b (xy) = log_b x + log_b y

Derivation of Quotient Formula of Log

Quotient formula of log states that,

log_b (x/y) = log_b x – log_b y

This is derived as,

Let take, log_b x = m and log_b y = n…(i)

Now using definition of logarithm,

x = b^m and y = bⁿ

⇒ x/y = b^m / bⁿ = b^{(m – n)} → {by a law of exponents, a^m / aⁿ = a^{(m – n)}}

Converting into logarithm form

m – n = logb (x/y)

from eq. (1)

log_b (x/y) = log_b x – log_b y

Derivation of Power Formula of Log

Power formula of log state that,

log_b a^x = x log_b a

This is derived as,

Let log_b a = m….(i)

Now using definition of logarithm,

a^x = (b^m)^x

⇒ a^x = (b^mx) {by a law of exponents, (a^m)ⁿ = a^mn}

Converting into logarithm form,

log_b a^x = m x

using eq. (i)

log_b a^x = x log_b a

Derivation of Change of Base Formula of Log

Change of base formula of log states that,

log_b a = (log_c a) / (log_c b)

This is derived as,

Let, log_b a = x, log_c a = y, and log_c b = z

In exponential forms,

a = b^x … (1)

a = c^y … (2)

b = c^z … (3)

From (1) and (2),

b^x = c^y

(c^z)^x = c^y (from (3))

⇒ c^zx = c^y

⇒ zx = y

⇒ x = y / z

Substituting values of x, y, and z back,

log_b a = (log_c a) / (log_c b)

Logarithm was invented in the 17th century by Scottish mathematician John Napier (1550-1617). The Napier logarithm was the first to be published in 1614. Henry Briggs introduced a common (base 10) logarithm. John Napier’s purpose was to assist in the multiplication of quantities that were called sines.