Logarithm Formulas and Their Proofs

Here we learn the general laws of logarithms. The list of logarithm formulas are given at the end of this post.

Proofs of Logarithm Formulas:

Proof of Product Rule of logarithms

log_a(MN)=log_a M + log_a N

Proof:

Suppose that x= log_a M

Thus, by the definition of logarithm

a^x=M

Again, suppose that y=log_a N  ⇒ a^y=N

Note that MN=a^x ⋅ a^y= a^x+y

Taking logarithms with respect to the base a on both sides,

log_a(MN)=log_a a^x+y = x+y $[\because \log_a a^k=k]$

= log_a M + log_a N (plug-in the values of x and y)

Hence, log_a(MN)=log_a M + log_a N (Proved)

In the same way as above, we can prove the quotient rule of logarithms.

Proof of Quotient Rule of Logarithms

log_a(M/N)=log_a M – log_a N

Proof:

Suppose that x=log_a M

Thus, by the definition of logarithm

a^x=M

Again, suppose that y=log_a N  ⇒ a^y=N

Note that M/N = $\dfrac{a^x}{a^y}$ =a^x-y

Taking logarithms with respect to the base $a$ on both sides,

log_a (M/N) = log_a a^x-y

= x-y $[\because \log_a a^k=k]$

= log_a M – log_a N (plug-in the values of x and y)

Hence, log_a(M/N)=log_a M – log_a N (Proved)

Proof of Power Rule of logarithms

$\log_a M^n=n \log_a M$

Proof:

Suppose that x=log_a Mⁿ

Thus, a^x=Mⁿ—-(i)

Again, let y=log_a M

This implies that a^y=M —-(ii)

Combining (i) and (ii) we get that

$a^x=M^n=(a^y)^n$ [$\because a^y=M$]

⇒ a^x=a^ny$

Equating the powers of a, we obtain that x=ny

Putting the values of x and y, we have

log_a Mⁿ=n log_a M (Proved)

Proof of Base Change Rule of logarithms

log_a M= log_b M × log_a b

Proof:

Let x=log_a M

Thus, a^x=M —-(i)

Now, assume that y=log_b M

⇒ b^y=M —(ii)

Again, assume z=log_a b

⇒ a^z=b —-(iii)

Combining (i) and (ii), we have

$a^x=b^y=(a^z)^b$ [$\because a^z=b$ by (iii)]

⇒ a^x=a^bz

Equating the powers of a, we obtain x=bz

Putting the values of x, b and z, we get that

log_a M= log_b M × log_a b  (Proved)

Corollary: 

log_b a = $\dfrac{1}{\log_a b}$

Proof:  Take M=a in the above base change formula. We have

log_a a= log_b a × log_a b

⇒ 1 = log_b a × log_a b [log_a a=1]

Hence, it follows that log_b a = $\dfrac{1}{\log_a b}$   (Proved)

Logarithm Formulas List

1.  log_a 1=0

2.  log_a a = 1

3.  $a^{\log_a M}=M$

4.  log_a(MN)=log_a M + log_a N

5.  log_a(M/N)=log_a M – log_a N

6.  log_a Mⁿ = n log_a M

7.  log_a M= log_b M × log_a b

8.  log_b a = 1/log_ab

Also read:

An Introduction to Logarithm

Logarithm Formulas with Proofs

Common logarithm and Natural Logarithm

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